fsuppind - Mathbox for Steven Nguyen

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Theorem fsuppind 43023

Description: Induction on functions 𝐹:𝐴⟶𝐵 with finite support, or in other words the base set of the free module (see frlmelbas 21736 and frlmplusgval 21744). This theorem is structurally general for polynomial proof usage (see mplelbas 21969 and mpladd 21987). 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 6854	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ V)
4		fsuppind.v	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5	3, 4	elmapd 8787	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (𝐵 ↑_m 𝐼) ↔ 𝑋:𝐼⟶𝐵))
6	5	adantr 480	. . . . . . 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 341	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 1 → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
9	8	ralbidv 3160	. . . . . . . . . . . . . 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 341	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
12	11	ralbidv 3160	. . . . . . . . . . . . . 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 341	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝑗 + 1) → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
15	14	ralbidv 3160	. . . . . . . . . . . . . 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 341	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
18	17	ralbidv 3160	. . . . . . . . . . . . . 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 7400	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ supp 0 ) ∈ V
21		euhash1 14382	. . . . . . . . . . . . . . . . . 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 275	. . . . . . . . . . . . . . . 16 ⊢ (1 = (♯‘(ℎ supp 0 )) ↔ ∃!𝑐 𝑐 ∈ (ℎ supp 0 ))
24		elmapfn 8812	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ (𝐵 ↑_m 𝐼) → ℎ Fn 𝐼)
25	24	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → ℎ Fn 𝐼)
26	4	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → 𝐼 ∈ 𝑉)
27		fsuppind.z	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 = (0_g‘𝐺)
28	27	fvexi 6854	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ V
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → 0 ∈ V)
30		elsuppfn 8120	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑐 ∈ (ℎ supp 0 ) ↔ (𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
31	25, 26, 29, 30	syl3anc 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (𝑐 ∈ (ℎ supp 0 ) ↔ (𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
32	31	eubidv 2586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 𝑐 ∈ (ℎ supp 0 ) ↔ ∃!𝑐(𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
33		df-reu 3343	. . . . . . . . . . . . . . . . . 18 ⊢ (∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ↔ ∃!𝑐(𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 ))
34	32, 33	bitr4di 289	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 𝑐 ∈ (ℎ supp 0 ) ↔ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ))
35	24	ad2antlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ Fn 𝐼)
36		fvex 6853	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ‘𝑥) ∈ V
37	36, 28	ifex 4517	. . . . . . . . . . . . . . . . . . . . . 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 6641	. . . . . . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑣 → (ℎ‘𝑥) = (ℎ‘𝑣))
43	41, 42	ifbieq1d 4491	. . . . . . . . . . . . . . . . . . . . . . 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 6953	. . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 = 𝑣 → (ℎ‘𝑐) = (ℎ‘𝑣))
50	49	neeq1d 2991	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 = 𝑣 → ((ℎ‘𝑐) ≠ 0 ↔ (ℎ‘𝑣) ≠ 0 ))
51	50	riota2 7349	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑣 ∈ 𝐼 ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ((ℎ‘𝑣) ≠ 0 ↔ (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) = 𝑣))
52	47, 48, 51	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . 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 314	. . . . . . . . . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . 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 4503	. . . . . . . . . . . . . . . . . . . . 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 6986	. . . . . . . . . . . . . . . . . . 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 7341	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 → (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∈ 𝐼)
63	62	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∈ 𝐼)
64		elmapi 8796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ (𝐵 ↑_m 𝐼) → ℎ:𝐼⟶𝐵)
65	64	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ:𝐼⟶𝐵)
66	65, 63	ffvelcdmd 7037	. . . . . . . . . . . . . . . . . . . 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 3180	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
69	68	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 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 4490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 ))
72	71	mpteq2dv 5179	. . . . . . . . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) ∧ 𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → 𝑏 = (ℎ‘𝑥))
77	76	ifeq1da 4498	. . . . . . . . . . . . . . . . . . . . . . 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 5179	. . . . . . . . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ ∈ 𝐻)
83	82	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 → ℎ ∈ 𝐻))
84	34, 83	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 𝑐 ∈ (ℎ supp 0 ) → ℎ ∈ 𝐻))
85	23, 84	biimtrid 242	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
86	85	ralrimiva 3129	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀ℎ ∈ (𝐵 ↑_m 𝐼)(1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
87		fvoveq1 7390	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → (♯‘(𝑚 supp 0 )) = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
88	87	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → (𝑗 = (♯‘(𝑚 supp 0 )) ↔ 𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 ))))
89		oveq1 7374	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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 ))))
90	89	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (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 )))))
91	88, 90	anbi12d 633	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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 ))))))
92		fsuppind.g	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐺 ∈ Grp)
93	1, 27	grpidcl 18941	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
94	92, 93	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 0 ∈ 𝐵)
95	94	ad5antr 735	. . . . . . . . . . . . . . . . . . . . . . . . 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)
96		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵 ↑_m 𝐼) = (𝐵 ↑_m 𝐼)
97		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))
98	97	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . 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))
99		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 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)
100	96, 98, 99	mapfvd 8827	. . . . . . . . . . . . . . . . . . . . . . . . 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)
101	95, 100	ifcld 4513	. . . . . . . . . . . . . . . . . . . . . . . 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)
102	101	fmpttd 7067	. . . . . . . . . . . . . . . . . . . . . . 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)
103	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)
104	4	ad4antr 733	. . . . . . . . . . . . . . . . . . . . . . . 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)
105	103, 104	elmapd 8787	. . . . . . . . . . . . . . . . . . . . . . 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))
106	102, 105	mpbird 257	. . . . . . . . . . . . . . . . . . . . . 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))
107	106	adantrl 717	. . . . . . . . . . . . . . . . . . . . 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))
108		ovexd 7402	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑙 supp 0 ) ∈ V)
109		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑧 ∈ 𝐼)
110		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑙‘𝑧) ≠ 0 )
111		elmapfn 8812	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑙 ∈ (𝐵 ↑_m 𝐼) → 𝑙 Fn 𝐼)
112	111	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 Fn 𝐼)
113	112	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑙 Fn 𝐼)
114	4	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝐼 ∈ 𝑉)
115	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 0 ∈ V)
116		elsuppfn 8120	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑧 ∈ (𝑙 supp 0 ) ↔ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )))
117	113, 114, 115, 116	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
118	109, 110, 117	mpbir2and 714	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑧 ∈ (𝑙 supp 0 ))
119		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 ∈ ℕ)
120	119	nnnn0d 12498	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 ∈ ℕ₀)
121		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 )))
122	121	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))
123		hashdifsnp1 14468	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑙 supp 0 ) ∈ V ∧ 𝑧 ∈ (𝑙 supp 0 ) ∧ 𝑗 ∈ ℕ₀) → ((♯‘(𝑙 supp 0 )) = (𝑗 + 1) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗))
124	123	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑙 supp 0 ) ∈ V ∧ 𝑧 ∈ (𝑙 supp 0 ) ∧ 𝑗 ∈ ℕ₀) ∧ (♯‘(𝑙 supp 0 )) = (𝑗 + 1)) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗)
125	108, 118, 120, 122, 124	syl31anc 1376	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗)
126		eldifsn 4731	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑣 ∈ ((𝑙 supp 0 ) ∖ {𝑧}) ↔ (𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣 ≠ 𝑧))
127		fvex 6853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑙‘𝑥) ∈ V
128	28, 127	ifex 4517	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) ∈ V
129		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))
130	128, 129	fnmpti 6641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) Fn 𝐼
131	130	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)
132	4	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝐼 ∈ 𝑉)
133	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 0 ∈ V)
134		elsuppfn 8120	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
135	131, 132, 133, 134	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
136		iftrue 4472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑣 = 𝑧 → if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 )
137		olc 869	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑣 = 𝑧 → ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧))
138	136, 137	2thd 265	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
139		iffalse 4475	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ 𝑣 = 𝑧 → if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = (𝑙‘𝑣))
140	139	eqeq1d 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ (𝑙‘𝑣) = 0 ))
141		biorf 937	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ 𝑣 = 𝑧 → ((𝑙‘𝑣) = 0 ↔ (𝑣 = 𝑧 ∨ (𝑙‘𝑣) = 0 )))
142		orcom 871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧) ↔ (𝑣 = 𝑧 ∨ (𝑙‘𝑣) = 0 ))
143	141, 142	bitr4di 289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑣 = 𝑧 → ((𝑙‘𝑣) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
144	140, 143	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (¬ 𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
145	138, 144	pm2.61i 182	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧))
146	145	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)))
147	146	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)))
148		neanior 3025	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑙‘𝑣) ≠ 0 ∧ 𝑣 ≠ 𝑧) ↔ ¬ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧))
149	147, 148	bitr4di 289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
150	149	anbi2d 631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
151		anass 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 ) ∧ 𝑣 ≠ 𝑧) ↔ (𝑣 ∈ 𝐼 ∧ ((𝑙‘𝑣) ≠ 0 ∧ 𝑣 ≠ 𝑧)))
152	150, 151	bitr4di 289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
153		equequ1 2027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑧 ↔ 𝑣 = 𝑧))
154		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = 𝑣 → (𝑙‘𝑥) = (𝑙‘𝑣))
155	153, 154	ifbieq2d 4493	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) = if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)))
156	155, 129, 128	fvmpt3i 6953	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑣 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) = if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)))
157	156	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
158	157	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 ))
159	158	pm5.32da 579	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
160	112	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝑙 Fn 𝐼)
161		elsuppfn 8120	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑣 ∈ (𝑙 supp 0 ) ↔ (𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 )))
162	160, 132, 133, 161	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
163	162	anbi1d 632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
164	152, 159, 163	3bitr4d 311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
165	135, 164	bitr2d 280	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
166	126, 165	bitrid 283	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
167	166	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 ))
168	167	fveq2d 6844	. . . . . . . . . . . . . . . . . . . . . . . . 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 )))
169	168	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . 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 )))
170	125, 169	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 )))
171	127, 28	ifex 4517	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ) ∈ V
172		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))
173	171, 172	fnmpti 6641	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) Fn 𝐼
174	173	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)
175		inidm 4167	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐼 ∩ 𝐼) = 𝐼
176	131, 174, 132, 132, 175	offn 7644	. . . . . . . . . . . . . . . . . . . . . . . . 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)
177	153, 154	ifbieq1d 4491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ) = if(𝑣 = 𝑧, (𝑙‘𝑣), 0 ))
178	177, 172, 171	fvmpt3i 6953	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑣 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))‘𝑣) = if(𝑣 = 𝑧, (𝑙‘𝑣), 0 ))
179	178	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))
180	131, 174, 132, 132, 175, 157, 179	ofval 7642	. . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
181	92	ad4antr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝐺 ∈ Grp)
182		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ ((𝑙‘𝑧) ≠ 0 ∧ 𝑣 ∈ 𝐼)) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
183	182	anassrs 467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
184		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝑣 ∈ 𝐼)
185	96, 183, 184	mapfvd 8827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (𝑙‘𝑣) ∈ 𝐵)
186		fsuppind.p	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ + = (+_g‘𝐺)
187	1, 186, 27	grplid 18943	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → ( 0 + (𝑙‘𝑣)) = (𝑙‘𝑣))
188	1, 186, 27	grprid 18944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → ((𝑙‘𝑣) + 0 ) = (𝑙‘𝑣))
189	187, 188	ifeq12d 4488	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → if(𝑣 = 𝑧, ( 0 + (𝑙‘𝑣)), ((𝑙‘𝑣) + 0 )) = if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣)))
190	181, 185, 189	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
191		ovif12 7467	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) + if(𝑣 = 𝑧, (𝑙‘𝑣), 0 )) = if(𝑣 = 𝑧, ( 0 + (𝑙‘𝑣)), ((𝑙‘𝑣) + 0 ))
192		ifid 4507	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣)) = (𝑙‘𝑣)
193	192	eqcomi 2745	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑙‘𝑣) = if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣))
194	190, 191, 193	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))
195	180, 194	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))
196	160, 176, 195	eqfnfvd 6986	. . . . . . . . . . . . . . . . . . . . . . . 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 ))))
197	196	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . 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 ))))
198	170, 197	jca 511	. . . . . . . . . . . . . . . . . . . . . 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 )))))
199	198	adantllr 720	. . . . . . . . . . . . . . . . . . . . 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 )))))
200	91, 107, 199	rspcedvdw 3567	. . . . . . . . . . . . . . . . . . . 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 )))))
201	111	ad2antrl 729	. . . . . . . . . . . . . . . . . . . . . . 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)
202	4	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . . . 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)
203	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)
204		suppvalfn 8118	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑙 supp 0 ) = {𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 })
205	201, 202, 203, 204	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . 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 })
206		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 )))
207		peano2nn 12186	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
208	207	ad3antlr 732	. . . . . . . . . . . . . . . . . . . . . . . . 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)
209	208	nnne0d 12227	. . . . . . . . . . . . . . . . . . . . . . . 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)
210	206, 209	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)
211		ovex 7400	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 supp 0 ) ∈ V
212		hasheq0 14325	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑙 supp 0 ) ∈ V → ((♯‘(𝑙 supp 0 )) = 0 ↔ (𝑙 supp 0 ) = ∅))
213	212	necon3bid 2976	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑙 supp 0 ) ∈ V → ((♯‘(𝑙 supp 0 )) ≠ 0 ↔ (𝑙 supp 0 ) ≠ ∅))
214	211, 213	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 ) ≠ ∅))
215	210, 214	mpbid 232	. . . . . . . . . . . . . . . . . . . . . 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 ) ≠ ∅)
216	205, 215	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 } ≠ ∅)
217		rabn0 4329	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 } ≠ ∅ ↔ ∃𝑧 ∈ 𝐼 (𝑙‘𝑧) ≠ 0 )
218	216, 217	sylib 218	. . . . . . . . . . . . . . . . . . . 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 )
219	200, 218	reximddv 3153	. . . . . . . . . . . . . . . . . . 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 )))))
220		rexcom 3266	. . . . . . . . . . . . . . . . . . 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 )))))
221	219, 220	sylib 218	. . . . . . . . . . . . . . . . . 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 )))))
222		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 ))))
223		fvoveq1 7390	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℎ = 𝑚 → (♯‘(ℎ supp 0 )) = (♯‘(𝑚 supp 0 )))
224	223	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑚 → (𝑗 = (♯‘(ℎ supp 0 )) ↔ 𝑗 = (♯‘(𝑚 supp 0 ))))
225		eleq1w 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑚 → (ℎ ∈ 𝐻 ↔ 𝑚 ∈ 𝐻))
226	224, 225	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ℎ = 𝑚 → ((𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚 ∈ 𝐻)))
227	226	rspccva 3563	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ∧ 𝑚 ∈ (𝐵 ↑_m 𝐼)) → (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚 ∈ 𝐻))
228	227	adantll 715	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ 𝑚 ∈ (𝐵 ↑_m 𝐼)) → (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚 ∈ 𝐻))
229	228	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((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	adantllr 720	. . . . . . . . . . . . . . . . . . . . . . . 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)
231	230	adantlrr 722	. . . . . . . . . . . . . . . . . . . . . . 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)
232	231	adantrr 718	. . . . . . . . . . . . . . . . . . . . . 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)
233		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)
234	97	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . 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))
235	96, 234, 233	mapfvd 8827	. . . . . . . . . . . . . . . . . . . . . . 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)
236	68	ad5antr 735	. . . . . . . . . . . . . . . . . . . . . . 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)
237		equequ2 2028	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = 𝑧 → (𝑥 = 𝑎 ↔ 𝑥 = 𝑧))
238	237	ifbid 4490	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑧 → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = 𝑧, 𝑏, 0 ))
239	238	mpteq2dv 5179	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑧 → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )))
240	239	eleq1d 2821	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑧 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) ∈ 𝐻))
241		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 𝑧 → (𝑙‘𝑥) = (𝑙‘𝑧))
242	241	eqeq2d 2747	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑧 → (𝑏 = (𝑙‘𝑥) ↔ 𝑏 = (𝑙‘𝑧)))
243	242	biimparc 479	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑏 = (𝑙‘𝑧) ∧ 𝑥 = 𝑧) → 𝑏 = (𝑙‘𝑥))
244	243	ifeq1da 4498	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 = (𝑙‘𝑧) → if(𝑥 = 𝑧, 𝑏, 0 ) = if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))
245	244	mpteq2dv 5179	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = (𝑙‘𝑧) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))
246	245	eleq1d 2821	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = (𝑙‘𝑧) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) ∈ 𝐻))
247	240, 246	rspc2va 3576	. . . . . . . . . . . . . . . . . . . . . . 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)
248	233, 235, 236, 247	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)
249		fsuppind.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 ∘_f + 𝑦) ∈ 𝐻)
250	249	ralrimivva 3180	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 ∘_f + 𝑦) ∈ 𝐻)
251	250	ad5antr 735	. . . . . . . . . . . . . . . . . . . . . 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)
252		ovrspc2v 7393	. . . . . . . . . . . . . . . . . . . . . 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)
253	232, 248, 251, 252	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)
254	222, 253	eqeltrd 2836	. . . . . . . . . . . . . . . . . . . 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)
255	254	ex 412	. . . . . . . . . . . . . . . . . . 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))
256	255	rexlimdvva 3194	. . . . . . . . . . . . . . . . . 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))
257	221, 256	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)
258	257	exp32 420	. . . . . . . . . . . . . . . 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)))
259	258	ralrimiv 3128	. . . . . . . . . . . . . . 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))
260		fvoveq1 7390	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = ℎ → (♯‘(𝑙 supp 0 )) = (♯‘(ℎ supp 0 )))
261	260	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = ℎ → ((𝑗 + 1) = (♯‘(𝑙 supp 0 )) ↔ (𝑗 + 1) = (♯‘(ℎ supp 0 ))))
262		eleq1w 2819	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = ℎ → (𝑙 ∈ 𝐻 ↔ ℎ ∈ 𝐻))
263	261, 262	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = ℎ → (((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙 ∈ 𝐻) ↔ ((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
264	263	cbvralvw 3215	. . . . . . . . . . . . . . 15 ⊢ (∀𝑙 ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙 ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
265	259, 264	sylib 218	. . . . . . . . . . . . . 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))
266	9, 12, 15, 18, 86, 265	nnindd 12194	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
267	266	ralrimiva 3129	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
268		ralcom 3265	. . . . . . . . . . . 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))
269	267, 268	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ∀ℎ ∈ (𝐵 ↑_m 𝐼)∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
270		biidd 262	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (♯‘(ℎ supp 0 )) → (ℎ ∈ 𝐻 ↔ ℎ ∈ 𝐻))
271	270	ceqsralv 3470	. . . . . . . . . . . . 13 ⊢ ((♯‘(ℎ supp 0 )) ∈ ℕ → (∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ℎ ∈ 𝐻))
272	271	biimpcd 249	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) → ((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻))
273	272	ralimi 3074	. . . . . . . . . . 11 ⊢ (∀ℎ ∈ (𝐵 ↑_m 𝐼)∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) → ∀ℎ ∈ (𝐵 ↑_m 𝐼)((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻))
274	269, 273	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∀ℎ ∈ (𝐵 ↑_m 𝐼)((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻))
275		fvoveq1 7390	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑋 → (♯‘(ℎ supp 0 )) = (♯‘(𝑋 supp 0 )))
276	275	eleq1d 2821	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑋 → ((♯‘(ℎ supp 0 )) ∈ ℕ ↔ (♯‘(𝑋 supp 0 )) ∈ ℕ))
277		eleq1 2824	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑋 → (ℎ ∈ 𝐻 ↔ 𝑋 ∈ 𝐻))
278	276, 277	imbi12d 344	. . . . . . . . . . 11 ⊢ (ℎ = 𝑋 → (((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻) ↔ ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋 ∈ 𝐻)))
279	278	rspcv 3560	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐵 ↑_m 𝐼) → (∀ℎ ∈ (𝐵 ↑_m 𝐼)((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻) → ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋 ∈ 𝐻)))
280	274, 279	syl5com 31	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (𝐵 ↑_m 𝐼) → ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋 ∈ 𝐻)))
281	280	com23 86	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(𝑋 supp 0 )) ∈ ℕ → (𝑋 ∈ (𝐵 ↑_m 𝐼) → 𝑋 ∈ 𝐻)))
282	281	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → (𝑋 ∈ (𝐵 ↑_m 𝐼) → 𝑋 ∈ 𝐻))
283	6, 282	sylbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → (𝑋:𝐼⟶𝐵 → 𝑋 ∈ 𝐻))
284	283	imp 406	. . . . 5 ⊢ (((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) ∧ 𝑋:𝐼⟶𝐵) → 𝑋 ∈ 𝐻)
285	284	an32s 653	. . . 4 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → 𝑋 ∈ 𝐻)
286	285	adantlr 716	. . 3 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → 𝑋 ∈ 𝐻)
287		ovex 7400	. . . . 5 ⊢ (𝑋 supp 0 ) ∈ V
288		hasheq0 14325	. . . . 5 ⊢ ((𝑋 supp 0 ) ∈ V → ((♯‘(𝑋 supp 0 )) = 0 ↔ (𝑋 supp 0 ) = ∅))
289	287, 288	ax-mp 5	. . . 4 ⊢ ((♯‘(𝑋 supp 0 )) = 0 ↔ (𝑋 supp 0 ) = ∅)
290		ffn 6668	. . . . . . . 8 ⊢ (𝑋:𝐼⟶𝐵 → 𝑋 Fn 𝐼)
291	290	ad2antlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 Fn 𝐼)
292	4	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝐼 ∈ 𝑉)
293	28	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 0 ∈ V)
294		fnsuppeq0 8142	. . . . . . 7 ⊢ ((𝑋 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → ((𝑋 supp 0 ) = ∅ ↔ 𝑋 = (𝐼 × { 0 })))
295	291, 292, 293, 294	syl3anc 1374	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → ((𝑋 supp 0 ) = ∅ ↔ 𝑋 = (𝐼 × { 0 })))
296	295	biimpa 476	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → 𝑋 = (𝐼 × { 0 }))
297		fsuppind.0	. . . . . 6 ⊢ (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
298	297	ad3antrrr 731	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → (𝐼 × { 0 }) ∈ 𝐻)
299	296, 298	eqeltrd 2836	. . . 4 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → 𝑋 ∈ 𝐻)
300	289, 299	sylan2b 595	. . 3 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (♯‘(𝑋 supp 0 )) = 0) → 𝑋 ∈ 𝐻)
301		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 finSupp 0 )
302	301	fsuppimpd 9282	. . . . 5 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → (𝑋 supp 0 ) ∈ Fin)
303		hashcl 14318	. . . . 5 ⊢ ((𝑋 supp 0 ) ∈ Fin → (♯‘(𝑋 supp 0 )) ∈ ℕ₀)
304	302, 303	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → (♯‘(𝑋 supp 0 )) ∈ ℕ₀)
305		elnn0 12439	. . . 4 ⊢ ((♯‘(𝑋 supp 0 )) ∈ ℕ₀ ↔ ((♯‘(𝑋 supp 0 )) ∈ ℕ ∨ (♯‘(𝑋 supp 0 )) = 0))
306	304, 305	sylib 218	. . 3 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → ((♯‘(𝑋 supp 0 )) ∈ ℕ ∨ (♯‘(𝑋 supp 0 )) = 0))
307	286, 300, 306	mpjaodan 961	. 2 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 ∈ 𝐻)
308	307	anasss 466	1 ⊢ ((𝜑 ∧ (𝑋:𝐼⟶𝐵 ∧ 𝑋 finSupp 0 )) → 𝑋 ∈ 𝐻)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∃!weu 2568 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 ∃!wreu 3340 {crab 3389 Vcvv 3429 ∖ cdif 3886 ∅c0 4273 ifcif 4466 {csn 4567 class class class wbr 5085 ↦ cmpt 5166 × cxp 5629 Fn wfn 6493 ⟶wf 6494 ‘cfv 6498 ℩crio 7323 (class class class)co 7367 ∘_f cof 7629 supp csupp 8110 ↑_m cmap 8773 Fincfn 8893 finSupp cfsupp 9274 0cc0 11038 1c1 11039 + caddc 11041 ℕcn 12174 ℕ₀cn0 12437 ♯chash 14292 Basecbs 17179 +_gcplusg 17220 0_gc0g 17402 Grpcgrp 18909

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-oadd 8409 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-hash 14293 df-0g 17404 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18912

This theorem is referenced by: fsuppssind 43026

W3C validator