fsuppind - Mathbox for Steven Nguyen

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Theorem fsuppind 39930

Description: Induction on functions 𝐹:𝐴⟶𝐵 with finite support, or in other words the base set of the free module (see frlmelbas 20672 and frlmplusgval 20680). This theorem is structurally general for polynomial proof usage (see mplelbas 20909 and mpladd 20923). Note that hypothesis 0 is redundant when 𝐼 is nonempty. (Contributed by SN, 18-May-2024.)

**Hypotheses**
Ref	Expression
fsuppind.b	⊢ 𝐵 = (Base‘𝐺)
fsuppind.z	⊢ 0 = (0_g‘𝐺)
fsuppind.p	⊢ + = (+_g‘𝐺)
fsuppind.g	⊢ (𝜑 → 𝐺 ∈ Grp)
fsuppind.v	⊢ (𝜑 → 𝐼 ∈ 𝑉)
fsuppind.0	⊢ (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
fsuppind.1	⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
fsuppind.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 ∘_f + 𝑦) ∈ 𝐻)

**Assertion**
Ref	Expression
fsuppind	⊢ ((𝜑 ∧ (𝑋:𝐼⟶𝐵 ∧ 𝑋 finSupp 0 )) → 𝑋 ∈ 𝐻)

Distinct variable groups: 𝑥, + ,𝑦 0 ,𝑎,𝑏,𝑥 𝑦, 0 𝐼,𝑎,𝑏,𝑥 𝑦,𝐼 𝐻,𝑏 𝑦,𝐻,𝑥 𝐻,𝑎 𝜑,𝑥,𝑦 𝜑,𝑎,𝑏 𝐵,𝑎,𝑏,𝑥 Allowed substitution hints: 𝐵(𝑦) + (𝑎,𝑏) 𝐺(𝑥,𝑦,𝑎,𝑏) 𝑉(𝑥,𝑦,𝑎,𝑏) 𝑋(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem fsuppind Dummy variables 𝑧 𝑐 ℎ 𝑚 𝑣 𝑖 𝑗 𝑛 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsuppind.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐺)
2	1	fvexi 6709	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ V)
4		fsuppind.v	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5	3, 4	elmapd 8500	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (𝐵 ↑_m 𝐼) ↔ 𝑋:𝐼⟶𝐵))
6	5	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → (𝑋 ∈ (𝐵 ↑_m 𝐼) ↔ 𝑋:𝐼⟶𝐵))
7		eqeq1 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 1 → (𝑖 = (♯‘(ℎ supp 0 )) ↔ 1 = (♯‘(ℎ supp 0 ))))
8	7	imbi1d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 1 → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
9	8	ralbidv 3108	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 1 → (∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
10		eqeq1 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → (𝑖 = (♯‘(ℎ supp 0 )) ↔ 𝑗 = (♯‘(ℎ supp 0 ))))
11	10	imbi1d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
12	11	ralbidv 3108	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
13		eqeq1 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝑗 + 1) → (𝑖 = (♯‘(ℎ supp 0 )) ↔ (𝑗 + 1) = (♯‘(ℎ supp 0 ))))
14	13	imbi1d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝑗 + 1) → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
15	14	ralbidv 3108	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑗 + 1) → (∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
16		eqeq1 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝑖 = (♯‘(ℎ supp 0 )) ↔ 𝑛 = (♯‘(ℎ supp 0 ))))
17	16	imbi1d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
18	17	ralbidv 3108	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑛 → (∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
19		eqcom 2743	. . . . . . . . . . . . . . . . 17 ⊢ (1 = (♯‘(ℎ supp 0 )) ↔ (♯‘(ℎ supp 0 )) = 1)
20		ovex 7224	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ supp 0 ) ∈ V
21		euhash1 13952	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ supp 0 ) ∈ V → ((♯‘(ℎ supp 0 )) = 1 ↔ ∃!𝑐 𝑐 ∈ (ℎ supp 0 )))
22	20, 21	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘(ℎ supp 0 )) = 1 ↔ ∃!𝑐 𝑐 ∈ (ℎ supp 0 ))
23	19, 22	bitri 278	. . . . . . . . . . . . . . . 16 ⊢ (1 = (♯‘(ℎ supp 0 )) ↔ ∃!𝑐 𝑐 ∈ (ℎ supp 0 ))
24		elmapfn 8524	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ (𝐵 ↑_m 𝐼) → ℎ Fn 𝐼)
25	24	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → ℎ Fn 𝐼)
26	4	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → 𝐼 ∈ 𝑉)
27		fsuppind.z	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 = (0_g‘𝐺)
28	27	fvexi 6709	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ V
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → 0 ∈ V)
30		elsuppfn 7891	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑐 ∈ (ℎ supp 0 ) ↔ (𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
31	25, 26, 29, 30	syl3anc 1373	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (𝑐 ∈ (ℎ supp 0 ) ↔ (𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
32	31	eubidv 2585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 𝑐 ∈ (ℎ supp 0 ) ↔ ∃!𝑐(𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
33		df-reu 3058	. . . . . . . . . . . . . . . . . 18 ⊢ (∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ↔ ∃!𝑐(𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 ))
34	32, 33	bitr4di 292	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 𝑐 ∈ (ℎ supp 0 ) ↔ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ))
35	24	ad2antlr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ Fn 𝐼)
36		fvex 6708	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ‘𝑥) ∈ V
37	36, 28	ifex 4475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ) ∈ V
38		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ))
39	37, 38	fnmpti 6499	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) Fn 𝐼
40	39	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) Fn 𝐼)
41		eqeq1 2740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑣 → (𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ↔ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
42		fveq2 6695	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑣 → (ℎ‘𝑥) = (ℎ‘𝑣))
43	41, 42	ifbieq1d 4449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑣 → if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ) = if(𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑣), 0 ))
44	43, 38, 37	fvmpt3i 6801	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ))‘𝑣) = if(𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑣), 0 ))
45	44	adantl 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ))‘𝑣) = if(𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑣), 0 ))
46		eqidd 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) ∧ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → (ℎ‘𝑣) = (ℎ‘𝑣))
47		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝑣 ∈ 𝐼)
48		simplr 769	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )
49		fveq2 6695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 = 𝑣 → (ℎ‘𝑐) = (ℎ‘𝑣))
50	49	neeq1d 2991	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 = 𝑣 → ((ℎ‘𝑐) ≠ 0 ↔ (ℎ‘𝑣) ≠ 0 ))
51	50	riota2 7174	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑣 ∈ 𝐼 ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ((ℎ‘𝑣) ≠ 0 ↔ (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) = 𝑣))
52	47, 48, 51	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ((ℎ‘𝑣) ≠ 0 ↔ (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) = 𝑣))
53		necom 2985	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ( 0 ≠ (ℎ‘𝑣) ↔ (ℎ‘𝑣) ≠ 0 )
54		eqcom 2743	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ↔ (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) = 𝑣)
55	52, 53, 54	3bitr4g 317	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ( 0 ≠ (ℎ‘𝑣) ↔ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
56	55	biimpd 232	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ( 0 ≠ (ℎ‘𝑣) → 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
57	56	necon1bd 2950	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (¬ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → 0 = (ℎ‘𝑣)))
58	57	imp 410	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) ∧ ¬ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → 0 = (ℎ‘𝑣))
59	46, 58	ifeqda 4461	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → if(𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑣), 0 ) = (ℎ‘𝑣))
60	45, 59	eqtr2d 2772	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (ℎ‘𝑣) = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ))‘𝑣))
61	35, 40, 60	eqfnfvd 6833	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ = (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )))
62		riotacl 7166	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 → (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∈ 𝐼)
63	62	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∈ 𝐼)
64		elmapi 8508	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ (𝐵 ↑_m 𝐼) → ℎ:𝐼⟶𝐵)
65	64	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ:𝐼⟶𝐵)
66	65, 63	ffvelrnd 6883	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) ∈ 𝐵)
67		fsuppind.1	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
68	67	ralrimivva 3102	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
69	68	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
70		eqeq2 2748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑥 = 𝑎 ↔ 𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
71	70	ifbid 4448	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 ))
72	71	mpteq2dv 5136	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 )))
73	72	eleq1d 2815	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 )) ∈ 𝐻))
74		fveq2 6695	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (ℎ‘𝑥) = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
75	74	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑏 = (ℎ‘𝑥) ↔ 𝑏 = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ))))
76	75	biimparc 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) ∧ 𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → 𝑏 = (ℎ‘𝑥))
77	76	ifeq1da 4456	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 ) = if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ))
78	77	mpteq2dv 5136	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )))
79	78	eleq1d 2815	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) ∈ 𝐻))
80	73, 79	rspc2va 3538	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∈ 𝐼 ∧ (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) ∈ 𝐻)
81	63, 66, 69, 80	syl21anc 838	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) ∈ 𝐻)
82	61, 81	eqeltrd 2831	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ ∈ 𝐻)
83	82	ex 416	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 → ℎ ∈ 𝐻))
84	34, 83	sylbid 243	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 𝑐 ∈ (ℎ supp 0 ) → ℎ ∈ 𝐻))
85	23, 84	syl5bi 245	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
86	85	ralrimiva 3095	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀ℎ ∈ (𝐵 ↑_m 𝐼)(1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
87		fsuppind.g	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐺 ∈ Grp)
88	1, 27	grpidcl 18349	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
89	87, 88	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 0 ∈ 𝐵)
90	89	ad5antr 734	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑥 ∈ 𝐼) → 0 ∈ 𝐵)
91		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵 ↑_m 𝐼) = (𝐵 ↑_m 𝐼)
92		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
93	92	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
94		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
95	91, 93, 94	mapfvd 8538	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑙‘𝑥) ∈ 𝐵)
96	90, 95	ifcld 4471	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑥 ∈ 𝐼) → if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) ∈ 𝐵)
97	96	fmpttd 6910	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))):𝐼⟶𝐵)
98	2	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝐵 ∈ V)
99	4	ad4antr 732	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝐼 ∈ 𝑉)
100	98, 99	elmapd 8500	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∈ (𝐵 ↑_m 𝐼) ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))):𝐼⟶𝐵))
101	97, 100	mpbird 260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∈ (𝐵 ↑_m 𝐼))
102	101	adantrl 716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∈ (𝐵 ↑_m 𝐼))
103		fvoveq1 7214	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → (♯‘(𝑚 supp 0 )) = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
104	103	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → (𝑗 = (♯‘(𝑚 supp 0 )) ↔ 𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 ))))
105		oveq1 7198	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))) = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))
106	105	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → (𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))) ↔ 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
107	104, 106	anbi12d 634	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → ((𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))) ↔ (𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )) ∧ 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))))
108	107	adantl 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) ∧ 𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))) → ((𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))) ↔ (𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )) ∧ 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))))
109		ovexd 7226	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑙 supp 0 ) ∈ V)
110		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑧 ∈ 𝐼)
111		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑙‘𝑧) ≠ 0 )
112		elmapfn 8524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑙 ∈ (𝐵 ↑_m 𝐼) → 𝑙 Fn 𝐼)
113	112	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 Fn 𝐼)
114	113	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑙 Fn 𝐼)
115	4	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝐼 ∈ 𝑉)
116	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 0 ∈ V)
117		elsuppfn 7891	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑧 ∈ (𝑙 supp 0 ) ↔ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )))
118	114, 115, 116, 117	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑧 ∈ (𝑙 supp 0 ) ↔ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )))
119	110, 111, 118	mpbir2and 713	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑧 ∈ (𝑙 supp 0 ))
120		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 ∈ ℕ)
121	120	nnnn0d 12115	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 ∈ ℕ₀)
122		simplrr 778	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑗 + 1) = (♯‘(𝑙 supp 0 )))
123	122	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (♯‘(𝑙 supp 0 )) = (𝑗 + 1))
124		hashdifsnp1 14027	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑙 supp 0 ) ∈ V ∧ 𝑧 ∈ (𝑙 supp 0 ) ∧ 𝑗 ∈ ℕ₀) → ((♯‘(𝑙 supp 0 )) = (𝑗 + 1) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗))
125	124	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑙 supp 0 ) ∈ V ∧ 𝑧 ∈ (𝑙 supp 0 ) ∧ 𝑗 ∈ ℕ₀) ∧ (♯‘(𝑙 supp 0 )) = (𝑗 + 1)) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗)
126	109, 119, 121, 123, 125	syl31anc 1375	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗)
127		eldifsn 4686	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑣 ∈ ((𝑙 supp 0 ) ∖ {𝑧}) ↔ (𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣 ≠ 𝑧))
128		fvex 6708	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑙‘𝑥) ∈ V
129	28, 128	ifex 4475	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) ∈ V
130		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))
131	129, 130	fnmpti 6499	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) Fn 𝐼
132	131	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) Fn 𝐼)
133	4	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝐼 ∈ 𝑉)
134	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 0 ∈ V)
135		elsuppfn 7891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑣 ∈ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 ) ↔ (𝑣 ∈ 𝐼 ∧ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) ≠ 0 )))
136	132, 133, 134, 135	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑣 ∈ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 ) ↔ (𝑣 ∈ 𝐼 ∧ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) ≠ 0 )))
137		iftrue 4431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑣 = 𝑧 → if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 )
138		olc 868	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑣 = 𝑧 → ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧))
139	137, 138	2thd 268	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
140		iffalse 4434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ 𝑣 = 𝑧 → if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = (𝑙‘𝑣))
141	140	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ (𝑙‘𝑣) = 0 ))
142		biorf 937	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ 𝑣 = 𝑧 → ((𝑙‘𝑣) = 0 ↔ (𝑣 = 𝑧 ∨ (𝑙‘𝑣) = 0 )))
143		orcom 870	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧) ↔ (𝑣 = 𝑧 ∨ (𝑙‘𝑣) = 0 ))
144	142, 143	bitr4di 292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑣 = 𝑧 → ((𝑙‘𝑣) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
145	141, 144	bitrd 282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (¬ 𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
146	139, 145	pm2.61i 185	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧))
147	146	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
148	147	necon3abid 2968	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 ↔ ¬ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
149		neanior 3024	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑙‘𝑣) ≠ 0 ∧ 𝑣 ≠ 𝑧) ↔ ¬ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧))
150	148, 149	bitr4di 292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 ↔ ((𝑙‘𝑣) ≠ 0 ∧ 𝑣 ≠ 𝑧)))
151	150	anbi2d 632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ 𝐼 ∧ if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 ) ↔ (𝑣 ∈ 𝐼 ∧ ((𝑙‘𝑣) ≠ 0 ∧ 𝑣 ≠ 𝑧))))
152		anass 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 ) ∧ 𝑣 ≠ 𝑧) ↔ (𝑣 ∈ 𝐼 ∧ ((𝑙‘𝑣) ≠ 0 ∧ 𝑣 ≠ 𝑧)))
153	151, 152	bitr4di 292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ 𝐼 ∧ if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 ) ↔ ((𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 ) ∧ 𝑣 ≠ 𝑧)))
154		equequ1 2035	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑧 ↔ 𝑣 = 𝑧))
155		fveq2 6695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = 𝑣 → (𝑙‘𝑥) = (𝑙‘𝑣))
156	154, 155	ifbieq2d 4451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) = if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)))
157	156, 130, 129	fvmpt3i 6801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑣 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) = if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)))
158	157	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) = if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)))
159	158	neeq1d 2991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) ≠ 0 ↔ if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 ))
160	159	pm5.32da 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ 𝐼 ∧ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) ≠ 0 ) ↔ (𝑣 ∈ 𝐼 ∧ if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 )))
161	113	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝑙 Fn 𝐼)
162		elsuppfn 7891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑣 ∈ (𝑙 supp 0 ) ↔ (𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 )))
163	161, 133, 134, 162	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑣 ∈ (𝑙 supp 0 ) ↔ (𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 )))
164	163	anbi1d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣 ≠ 𝑧) ↔ ((𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 ) ∧ 𝑣 ≠ 𝑧)))
165	153, 160, 164	3bitr4d 314	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ 𝐼 ∧ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) ≠ 0 ) ↔ (𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣 ≠ 𝑧)))
166	136, 165	bitr2d 283	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣 ≠ 𝑧) ↔ 𝑣 ∈ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
167	127, 166	syl5bb 286	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑣 ∈ ((𝑙 supp 0 ) ∖ {𝑧}) ↔ 𝑣 ∈ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
168	167	eqrdv 2734	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑙 supp 0 ) ∖ {𝑧}) = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 ))
169	168	fveq2d 6699	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
170	169	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
171	126, 170	eqtr3d 2773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
172	128, 28	ifex 4475	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ) ∈ V
173		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))
174	172, 173	fnmpti 6499	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) Fn 𝐼
175	174	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) Fn 𝐼)
176		inidm 4119	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐼 ∩ 𝐼) = 𝐼
177	132, 175, 133, 133, 176	offn 7459	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))) Fn 𝐼)
178	154, 155	ifbieq1d 4449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ) = if(𝑣 = 𝑧, (𝑙‘𝑣), 0 ))
179	178, 173, 172	fvmpt3i 6801	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑣 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))‘𝑣) = if(𝑣 = 𝑧, (𝑙‘𝑣), 0 ))
180	179	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))‘𝑣) = if(𝑣 = 𝑧, (𝑙‘𝑣), 0 ))
181	132, 175, 133, 133, 176, 158, 180	ofval 7457	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))‘𝑣) = (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) + if(𝑣 = 𝑧, (𝑙‘𝑣), 0 )))
182	87	ad4antr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝐺 ∈ Grp)
183		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ ((𝑙‘𝑧) ≠ 0 ∧ 𝑣 ∈ 𝐼)) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
184	183	anassrs 471	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
185		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝑣 ∈ 𝐼)
186	91, 184, 185	mapfvd 8538	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (𝑙‘𝑣) ∈ 𝐵)
187		fsuppind.p	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ + = (+_g‘𝐺)
188	1, 187, 27	grplid 18351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → ( 0 + (𝑙‘𝑣)) = (𝑙‘𝑣))
189	1, 187, 27	grprid 18352	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → ((𝑙‘𝑣) + 0 ) = (𝑙‘𝑣))
190	188, 189	ifeq12d 4446	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → if(𝑣 = 𝑧, ( 0 + (𝑙‘𝑣)), ((𝑙‘𝑣) + 0 )) = if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣)))
191	182, 186, 190	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → if(𝑣 = 𝑧, ( 0 + (𝑙‘𝑣)), ((𝑙‘𝑣) + 0 )) = if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣)))
192		ovif12 7288	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) + if(𝑣 = 𝑧, (𝑙‘𝑣), 0 )) = if(𝑣 = 𝑧, ( 0 + (𝑙‘𝑣)), ((𝑙‘𝑣) + 0 ))
193		ifid 4465	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣)) = (𝑙‘𝑣)
194	193	eqcomi 2745	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑙‘𝑣) = if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣))
195	191, 192, 194	3eqtr4g 2796	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) + if(𝑣 = 𝑧, (𝑙‘𝑣), 0 )) = (𝑙‘𝑣))
196	181, 195	eqtr2d 2772	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (𝑙‘𝑣) = (((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))‘𝑣))
197	161, 177, 196	eqfnfvd 6833	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))
198	197	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))
199	171, 198	jca 515	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )) ∧ 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
200	199	adantllr 719	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )) ∧ 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
201	102, 108, 200	rspcedvd 3530	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → ∃𝑚 ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
202	112	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 Fn 𝐼)
203	4	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝐼 ∈ 𝑉)
204	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 0 ∈ V)
205		suppvalfn 7889	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑙 supp 0 ) = {𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 })
206	202, 203, 204, 205	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑙 supp 0 ) = {𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 })
207		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑗 + 1) = (♯‘(𝑙 supp 0 )))
208		peano2nn 11807	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
209	208	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑗 + 1) ∈ ℕ)
210	209	nnne0d 11845	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑗 + 1) ≠ 0)
211	207, 210	eqnetrrd 3000	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (♯‘(𝑙 supp 0 )) ≠ 0)
212		ovex 7224	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 supp 0 ) ∈ V
213		hasheq0 13895	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑙 supp 0 ) ∈ V → ((♯‘(𝑙 supp 0 )) = 0 ↔ (𝑙 supp 0 ) = ∅))
214	213	necon3bid 2976	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑙 supp 0 ) ∈ V → ((♯‘(𝑙 supp 0 )) ≠ 0 ↔ (𝑙 supp 0 ) ≠ ∅))
215	212, 214	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ((♯‘(𝑙 supp 0 )) ≠ 0 ↔ (𝑙 supp 0 ) ≠ ∅))
216	211, 215	mpbid 235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑙 supp 0 ) ≠ ∅)
217	206, 216	eqnetrrd 3000	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → {𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 } ≠ ∅)
218		rabn0 4286	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 } ≠ ∅ ↔ ∃𝑧 ∈ 𝐼 (𝑙‘𝑧) ≠ 0 )
219	217, 218	sylib 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ∃𝑧 ∈ 𝐼 (𝑙‘𝑧) ≠ 0 )
220	201, 219	reximddv 3184	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ∃𝑧 ∈ 𝐼 ∃𝑚 ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
221		rexcom 3258	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑧 ∈ 𝐼 ∃𝑚 ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))) ↔ ∃𝑚 ∈ (𝐵 ↑_m 𝐼)∃𝑧 ∈ 𝐼 (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
222	220, 221	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ∃𝑚 ∈ (𝐵 ↑_m 𝐼)∃𝑧 ∈ 𝐼 (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
223		simprr 773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))
224		fvoveq1 7214	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℎ = 𝑚 → (♯‘(ℎ supp 0 )) = (♯‘(𝑚 supp 0 )))
225	224	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑚 → (𝑗 = (♯‘(ℎ supp 0 )) ↔ 𝑗 = (♯‘(𝑚 supp 0 ))))
226		eleq1w 2813	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑚 → (ℎ ∈ 𝐻 ↔ 𝑚 ∈ 𝐻))
227	225, 226	imbi12d 348	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ℎ = 𝑚 → ((𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚 ∈ 𝐻)))
228	227	rspccva 3526	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ∧ 𝑚 ∈ (𝐵 ↑_m 𝐼)) → (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚 ∈ 𝐻))
229	228	adantll 714	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ 𝑚 ∈ (𝐵 ↑_m 𝐼)) → (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚 ∈ 𝐻))
230	229	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ 𝑚 ∈ (𝐵 ↑_m 𝐼)) ∧ 𝑗 = (♯‘(𝑚 supp 0 ))) → 𝑚 ∈ 𝐻)
231	230	adantllr 719	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ 𝑚 ∈ (𝐵 ↑_m 𝐼)) ∧ 𝑗 = (♯‘(𝑚 supp 0 ))) → 𝑚 ∈ 𝐻)
232	231	adantlrr 721	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ 𝑗 = (♯‘(𝑚 supp 0 ))) → 𝑚 ∈ 𝐻)
233	232	adantrr 717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → 𝑚 ∈ 𝐻)
234		simplrr 778	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → 𝑧 ∈ 𝐼)
235	92	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
236	91, 235, 234	mapfvd 8538	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → (𝑙‘𝑧) ∈ 𝐵)
237	68	ad5antr 734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
238		equequ2 2036	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = 𝑧 → (𝑥 = 𝑎 ↔ 𝑥 = 𝑧))
239	238	ifbid 4448	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑧 → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = 𝑧, 𝑏, 0 ))
240	239	mpteq2dv 5136	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑧 → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )))
241	240	eleq1d 2815	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑧 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) ∈ 𝐻))
242		fveq2 6695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 𝑧 → (𝑙‘𝑥) = (𝑙‘𝑧))
243	242	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑧 → (𝑏 = (𝑙‘𝑥) ↔ 𝑏 = (𝑙‘𝑧)))
244	243	biimparc 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑏 = (𝑙‘𝑧) ∧ 𝑥 = 𝑧) → 𝑏 = (𝑙‘𝑥))
245	244	ifeq1da 4456	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 = (𝑙‘𝑧) → if(𝑥 = 𝑧, 𝑏, 0 ) = if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))
246	245	mpteq2dv 5136	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = (𝑙‘𝑧) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))
247	246	eleq1d 2815	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = (𝑙‘𝑧) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) ∈ 𝐻))
248	241, 247	rspc2va 3538	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) ∈ 𝐻)
249	234, 236, 237, 248	syl21anc 838	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) ∈ 𝐻)
250		fsuppind.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 ∘_f + 𝑦) ∈ 𝐻)
251	250	ralrimivva 3102	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 ∘_f + 𝑦) ∈ 𝐻)
252	251	ad5antr 734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 ∘_f + 𝑦) ∈ 𝐻)
253		ovrspc2v 7217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑚 ∈ 𝐻 ∧ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) ∈ 𝐻) ∧ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 ∘_f + 𝑦) ∈ 𝐻) → (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))) ∈ 𝐻)
254	233, 249, 252, 253	syl21anc 838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))) ∈ 𝐻)
255	223, 254	eqeltrd 2831	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → 𝑙 ∈ 𝐻)
256	255	ex 416	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) → ((𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))) → 𝑙 ∈ 𝐻))
257	256	rexlimdvva 3203	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (∃𝑚 ∈ (𝐵 ↑_m 𝐼)∃𝑧 ∈ 𝐼 (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))) → 𝑙 ∈ 𝐻))
258	222, 257	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 ∈ 𝐻)
259	258	exp32 424	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) → (𝑙 ∈ (𝐵 ↑_m 𝐼) → ((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙 ∈ 𝐻)))
260	259	ralrimiv 3094	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) → ∀𝑙 ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙 ∈ 𝐻))
261		fvoveq1 7214	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = ℎ → (♯‘(𝑙 supp 0 )) = (♯‘(ℎ supp 0 )))
262	261	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = ℎ → ((𝑗 + 1) = (♯‘(𝑙 supp 0 )) ↔ (𝑗 + 1) = (♯‘(ℎ supp 0 ))))
263		eleq1w 2813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = ℎ → (𝑙 ∈ 𝐻 ↔ ℎ ∈ 𝐻))
264	262, 263	imbi12d 348	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = ℎ → (((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙 ∈ 𝐻) ↔ ((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
265	264	cbvralvw 3348	. . . . . . . . . . . . . . 15 ⊢ (∀𝑙 ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙 ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
266	260, 265	sylib 221	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) → ∀ℎ ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
267	9, 12, 15, 18, 86, 266	nnindd 11815	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
268	267	ralrimiva 3095	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
269		ralcom 3257	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
270	268, 269	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → ∀ℎ ∈ (𝐵 ↑_m 𝐼)∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
271		biidd 265	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (♯‘(ℎ supp 0 )) → (ℎ ∈ 𝐻 ↔ ℎ ∈ 𝐻))
272	271	ceqsralv 3435	. . . . . . . . . . . . 13 ⊢ ((♯‘(ℎ supp 0 )) ∈ ℕ → (∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ℎ ∈ 𝐻))
273	272	biimpcd 252	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) → ((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻))
274	273	ralimi 3073	. . . . . . . . . . 11 ⊢ (∀ℎ ∈ (𝐵 ↑_m 𝐼)∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) → ∀ℎ ∈ (𝐵 ↑_m 𝐼)((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻))
275	270, 274	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∀ℎ ∈ (𝐵 ↑_m 𝐼)((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻))
276		fvoveq1 7214	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑋 → (♯‘(ℎ supp 0 )) = (♯‘(𝑋 supp 0 )))
277	276	eleq1d 2815	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑋 → ((♯‘(ℎ supp 0 )) ∈ ℕ ↔ (♯‘(𝑋 supp 0 )) ∈ ℕ))
278		eleq1 2818	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑋 → (ℎ ∈ 𝐻 ↔ 𝑋 ∈ 𝐻))
279	277, 278	imbi12d 348	. . . . . . . . . . 11 ⊢ (ℎ = 𝑋 → (((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻) ↔ ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋 ∈ 𝐻)))
280	279	rspcv 3522	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐵 ↑_m 𝐼) → (∀ℎ ∈ (𝐵 ↑_m 𝐼)((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻) → ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋 ∈ 𝐻)))
281	275, 280	syl5com 31	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (𝐵 ↑_m 𝐼) → ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋 ∈ 𝐻)))
282	281	com23 86	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(𝑋 supp 0 )) ∈ ℕ → (𝑋 ∈ (𝐵 ↑_m 𝐼) → 𝑋 ∈ 𝐻)))
283	282	imp 410	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → (𝑋 ∈ (𝐵 ↑_m 𝐼) → 𝑋 ∈ 𝐻))
284	6, 283	sylbird 263	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → (𝑋:𝐼⟶𝐵 → 𝑋 ∈ 𝐻))
285	284	imp 410	. . . . 5 ⊢ (((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) ∧ 𝑋:𝐼⟶𝐵) → 𝑋 ∈ 𝐻)
286	285	an32s 652	. . . 4 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → 𝑋 ∈ 𝐻)
287	286	adantlr 715	. . 3 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → 𝑋 ∈ 𝐻)
288		ovex 7224	. . . . 5 ⊢ (𝑋 supp 0 ) ∈ V
289		hasheq0 13895	. . . . 5 ⊢ ((𝑋 supp 0 ) ∈ V → ((♯‘(𝑋 supp 0 )) = 0 ↔ (𝑋 supp 0 ) = ∅))
290	288, 289	ax-mp 5	. . . 4 ⊢ ((♯‘(𝑋 supp 0 )) = 0 ↔ (𝑋 supp 0 ) = ∅)
291		ffn 6523	. . . . . . . 8 ⊢ (𝑋:𝐼⟶𝐵 → 𝑋 Fn 𝐼)
292	291	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 Fn 𝐼)
293	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝐼 ∈ 𝑉)
294	28	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 0 ∈ V)
295		fnsuppeq0 7912	. . . . . . 7 ⊢ ((𝑋 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → ((𝑋 supp 0 ) = ∅ ↔ 𝑋 = (𝐼 × { 0 })))
296	292, 293, 294, 295	syl3anc 1373	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → ((𝑋 supp 0 ) = ∅ ↔ 𝑋 = (𝐼 × { 0 })))
297	296	biimpa 480	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → 𝑋 = (𝐼 × { 0 }))
298		fsuppind.0	. . . . . 6 ⊢ (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
299	298	ad3antrrr 730	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → (𝐼 × { 0 }) ∈ 𝐻)
300	297, 299	eqeltrd 2831	. . . 4 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → 𝑋 ∈ 𝐻)
301	290, 300	sylan2b 597	. . 3 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (♯‘(𝑋 supp 0 )) = 0) → 𝑋 ∈ 𝐻)
302		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 finSupp 0 )
303	302	fsuppimpd 8970	. . . . 5 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → (𝑋 supp 0 ) ∈ Fin)
304		hashcl 13888	. . . . 5 ⊢ ((𝑋 supp 0 ) ∈ Fin → (♯‘(𝑋 supp 0 )) ∈ ℕ₀)
305	303, 304	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → (♯‘(𝑋 supp 0 )) ∈ ℕ₀)
306		elnn0 12057	. . . 4 ⊢ ((♯‘(𝑋 supp 0 )) ∈ ℕ₀ ↔ ((♯‘(𝑋 supp 0 )) ∈ ℕ ∨ (♯‘(𝑋 supp 0 )) = 0))
307	305, 306	sylib 221	. . 3 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → ((♯‘(𝑋 supp 0 )) ∈ ℕ ∨ (♯‘(𝑋 supp 0 )) = 0))
308	287, 301, 307	mpjaodan 959	. 2 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 ∈ 𝐻)
309	308	anasss 470	1 ⊢ ((𝜑 ∧ (𝑋:𝐼⟶𝐵 ∧ 𝑋 finSupp 0 )) → 𝑋 ∈ 𝐻)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∃!weu 2567 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 ∃!wreu 3053 {crab 3055 Vcvv 3398 ∖ cdif 3850 ∅c0 4223 ifcif 4425 {csn 4527 class class class wbr 5039 ↦ cmpt 5120 × cxp 5534 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 ℩crio 7147 (class class class)co 7191 ∘_f cof 7445 supp csupp 7881 ↑_m cmap 8486 Fincfn 8604 finSupp cfsupp 8963 0cc0 10694 1c1 10695 + caddc 10697 ℕcn 11795 ℕ₀cn0 12055 ♯chash 13861 Basecbs 16666 +_gcplusg 16749 0_gc0g 16898 Grpcgrp 18319

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-hash 13862 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322

This theorem is referenced by: fsuppssind 39933

W3C validator