fsuppind - Mathbox for Steven Nguyen

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Theorem fsuppind 42613

Description: Induction on functions 𝐹:𝐴⟶𝐵 with finite support, or in other words the base set of the free module (see frlmelbas 21716 and frlmplusgval 21724). This theorem is structurally general for polynomial proof usage (see mplelbas 21951 and mpladd 21969). 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 6890	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ V)
4		fsuppind.v	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5	3, 4	elmapd 8854	. . . . . . . 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 2739	. . . . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . 17 ⊢ (1 = (♯‘(ℎ supp 0 )) ↔ (♯‘(ℎ supp 0 )) = 1)
20		ovex 7438	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ supp 0 ) ∈ V
21		euhash1 14438	. . . . . . . . . . . . . . . . . 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 8879	. . . . . . . . . . . . . . . . . . . . 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 6890	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ V
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → 0 ∈ V)
30		elsuppfn 8169	. . . . . . . . . . . . . . . . . . . 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 3360	. . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ Fn 𝐼)
36		fvex 6889	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ‘𝑥) ∈ V
37	36, 28	ifex 4551	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ) ∈ V
38		eqid 2735	. . . . . . . . . . . . . . . . . . . . . 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 6681	. . . . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑣 → (𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ↔ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
42		fveq2 6876	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑣 → (ℎ‘𝑥) = (ℎ‘𝑣))
43	41, 42	ifbieq1d 4525	. . . . . . . . . . . . . . . . . . . . . . 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 6991	. . . . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )
49		fveq2 6876	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 = 𝑣 → (ℎ‘𝑐) = (ℎ‘𝑣))
50	49	neeq1d 2991	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 = 𝑣 → ((ℎ‘𝑐) ≠ 0 ↔ (ℎ‘𝑣) ≠ 0 ))
51	50	riota2 7387	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑣 ∈ 𝐼 ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ((ℎ‘𝑣) ≠ 0 ↔ (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) = 𝑣))
52	47, 48, 51	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4537	. . . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . . 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 7024	. . . . . . . . . . . . . . . . . . 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 7379	. . . . . . . . . . . . . . . . . . . . 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 8863	. . . . . . . . . . . . . . . . . . . . . 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	ffvelcdmd 7075	. . . . . . . . . . . . . . . . . . . 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 3187	. . . . . . . . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑥 = 𝑎 ↔ 𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
71	70	ifbid 4524	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 ))
72	71	mpteq2dv 5215	. . . . . . . . . . . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . . . . . . . . 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 6876	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (ℎ‘𝑥) = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
75	74	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . 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 4532	. . . . . . . . . . . . . . . . . . . . . . 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 5215	. . . . . . . . . . . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . . . . . . . . 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 3613	. . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . . 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 3132	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀ℎ ∈ (𝐵 ↑_m 𝐼)(1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
87		fvoveq1 7428	. . . . . . . . . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . . . . . . . . . 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 7412	. . . . . . . . . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . . . . . . . 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 18948	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
94	92, 93	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 0 ∈ 𝐵)
95	94	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)
96		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵 ↑_m 𝐼) = (𝐵 ↑_m 𝐼)
97		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 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))
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 8893	. . . . . . . . . . . . . . . . . . . . . . . . 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 4547	. . . . . . . . . . . . . . . . . . . . . . . 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 7105	. . . . . . . . . . . . . . . . . . . . . . 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 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)
105	103, 104	elmapd 8854	. . . . . . . . . . . . . . . . . . . . . . 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 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))
108		ovexd 7440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑙 supp 0 ) ∈ V)
109		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑧 ∈ 𝐼)
110		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑙‘𝑧) ≠ 0 )
111		elmapfn 8879	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑙 ∈ (𝐵 ↑_m 𝐼) → 𝑙 Fn 𝐼)
112	111	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 8169	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑧 ∈ (𝑙 supp 0 ) ↔ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )))
117	113, 114, 115, 116	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 )))
118	109, 110, 117	mpbir2and 713	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑧 ∈ (𝑙 supp 0 ))
119		simpllr 775	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 ∈ ℕ)
120	119	nnnn0d 12562	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 ∈ ℕ₀)
121		simplrr 777	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . . . . . . . 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 14524	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . . . . . . . . . . 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 4762	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑣 ∈ ((𝑙 supp 0 ) ∖ {𝑧}) ↔ (𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣 ≠ 𝑧))
127		fvex 6889	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑙‘𝑥) ∈ V
128	28, 127	ifex 4551	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) ∈ V
129		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))
130	128, 129	fnmpti 6681	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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 )))
136		iftrue 4506	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑣 = 𝑧 → if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 )
137		olc 868	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4509	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ 𝑣 = 𝑧 → if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = (𝑙‘𝑣))
140	139	eqeq1d 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ (𝑙‘𝑣) = 0 ))
141		biorf 936	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ 𝑣 = 𝑧 → ((𝑙‘𝑣) = 0 ↔ (𝑣 = 𝑧 ∨ (𝑙‘𝑣) = 0 )))
142		orcom 870	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2024	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑧 ↔ 𝑣 = 𝑧))
154		fveq2 6876	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = 𝑣 → (𝑙‘𝑥) = (𝑙‘𝑣))
155	153, 154	ifbieq2d 4527	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) = if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)))
156	155, 129, 128	fvmpt3i 6991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8169	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑣 ∈ (𝑙 supp 0 ) ↔ (𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 )))
162	160, 132, 133, 161	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 )))
163	162	anbi1d 631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6880	. . . . . . . . . . . . . . . . . . . . . . . . 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 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 )))
170	125, 169	eqtr3d 2772	. . . . . . . . . . . . . . . . . . . . . . 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 4551	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ) ∈ V
172		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))
173	171, 172	fnmpti 6681	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4202	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐼 ∩ 𝐼) = 𝐼
176	131, 174, 132, 132, 175	offn 7684	. . . . . . . . . . . . . . . . . . . . . . . . 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 4525	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ) = if(𝑣 = 𝑧, (𝑙‘𝑣), 0 ))
178	177, 172, 171	fvmpt3i 6991	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7682	. . . . . . . . . . . . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝐺 ∈ Grp)
182		simplrl 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8893	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18950	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → ( 0 + (𝑙‘𝑣)) = (𝑙‘𝑣))
188	1, 186, 27	grprid 18951	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → ((𝑙‘𝑣) + 0 ) = (𝑙‘𝑣))
189	187, 188	ifeq12d 4522	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7507	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) + if(𝑣 = 𝑧, (𝑙‘𝑣), 0 )) = if(𝑣 = 𝑧, ( 0 + (𝑙‘𝑣)), ((𝑙‘𝑣) + 0 ))
192		ifid 4541	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣)) = (𝑙‘𝑣)
193	192	eqcomi 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑙‘𝑣) = if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣))
194	190, 191, 193	3eqtr4g 2795	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . . . . . . . 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 7024	. . . . . . . . . . . . . . . . . . . . . . . 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 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 ))))
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 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 )))))
200	91, 107, 199	rspcedvdw 3604	. . . . . . . . . . . . . . . . . . . 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 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)
202	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)
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 8167	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑙 supp 0 ) = {𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 })
205	201, 202, 203, 204	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 })
206		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . 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 12252	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
208	207	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)
209	208	nnne0d 12290	. . . . . . . . . . . . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 supp 0 ) ∈ V
212		hasheq0 14381	. . . . . . . . . . . . . . . . . . . . . . . . 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 4364	. . . . . . . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . . . . . . 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 3271	. . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℎ = 𝑚 → (♯‘(ℎ supp 0 )) = (♯‘(𝑚 supp 0 )))
224	223	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑚 → (𝑗 = (♯‘(ℎ supp 0 )) ↔ 𝑗 = (♯‘(𝑚 supp 0 ))))
225		eleq1w 2817	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3600	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ∧ 𝑚 ∈ (𝐵 ↑_m 𝐼)) → (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚 ∈ 𝐻))
228	227	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))
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 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)
231	230	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)
232	231	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)
233		simplrr 777	. . . . . . . . . . . . . . . . . . . . . . 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 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))
235	96, 234, 233	mapfvd 8893	. . . . . . . . . . . . . . . . . . . . . . 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 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)
237		equequ2 2025	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = 𝑧 → (𝑥 = 𝑎 ↔ 𝑥 = 𝑧))
238	237	ifbid 4524	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑧 → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = 𝑧, 𝑏, 0 ))
239	238	mpteq2dv 5215	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑧 → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )))
240	239	eleq1d 2819	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑧 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) ∈ 𝐻))
241		fveq2 6876	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 𝑧 → (𝑙‘𝑥) = (𝑙‘𝑧))
242	241	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4532	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 = (𝑙‘𝑧) → if(𝑥 = 𝑧, 𝑏, 0 ) = if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))
245	244	mpteq2dv 5215	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = (𝑙‘𝑧) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))
246	245	eleq1d 2819	. . . . . . . . . . . . . . . . . . . . . . . 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 3613	. . . . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . . . 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 3187	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 ∘_f + 𝑦) ∈ 𝐻)
251	250	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)
252		ovrspc2v 7431	. . . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . . . . 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 3198	. . . . . . . . . . . . . . . . . 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 3131	. . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = ℎ → (♯‘(𝑙 supp 0 )) = (♯‘(ℎ supp 0 )))
261	260	eqeq2d 2746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = ℎ → ((𝑗 + 1) = (♯‘(𝑙 supp 0 )) ↔ (𝑗 + 1) = (♯‘(ℎ supp 0 ))))
262		eleq1w 2817	. . . . . . . . . . . . . . . . 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 3220	. . . . . . . . . . . . . . 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 12260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
267	266	ralrimiva 3132	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
268		ralcom 3270	. . . . . . . . . . . 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 3501	. . . . . . . . . . . . 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 3073	. . . . . . . . . . 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 7428	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑋 → (♯‘(ℎ supp 0 )) = (♯‘(𝑋 supp 0 )))
276	275	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑋 → ((♯‘(ℎ supp 0 )) ∈ ℕ ↔ (♯‘(𝑋 supp 0 )) ∈ ℕ))
277		eleq1 2822	. . . . . . . . . . . 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 3597	. . . . . . . . . 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 652	. . . 4 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → 𝑋 ∈ 𝐻)
286	285	adantlr 715	. . 3 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → 𝑋 ∈ 𝐻)
287		ovex 7438	. . . . 5 ⊢ (𝑋 supp 0 ) ∈ V
288		hasheq0 14381	. . . . 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 6706	. . . . . . . 8 ⊢ (𝑋:𝐼⟶𝐵 → 𝑋 Fn 𝐼)
291	290	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 Fn 𝐼)
292	4	ad2antrr 726	. . . . . . 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 8191	. . . . . . 7 ⊢ ((𝑋 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → ((𝑋 supp 0 ) = ∅ ↔ 𝑋 = (𝐼 × { 0 })))
295	291, 292, 293, 294	syl3anc 1373	. . . . . 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 730	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → (𝐼 × { 0 }) ∈ 𝐻)
299	296, 298	eqeltrd 2834	. . . 4 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → 𝑋 ∈ 𝐻)
300	289, 299	sylan2b 594	. . 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 9381	. . . . 5 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → (𝑋 supp 0 ) ∈ Fin)
303		hashcl 14374	. . . . 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 12503	. . . 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 960	. 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 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∃!weu 2567 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 ∃!wreu 3357 {crab 3415 Vcvv 3459 ∖ cdif 3923 ∅c0 4308 ifcif 4500 {csn 4601 class class class wbr 5119 ↦ cmpt 5201 × cxp 5652 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 ℩crio 7361 (class class class)co 7405 ∘_f cof 7669 supp csupp 8159 ↑_m cmap 8840 Fincfn 8959 finSupp cfsupp 9373 0cc0 11129 1c1 11130 + caddc 11132 ℕcn 12240 ℕ₀cn0 12501 ♯chash 14348 Basecbs 17228 +_gcplusg 17271 0_gc0g 17453 Grpcgrp 18916

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-hash 14349 df-0g 17455 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-grp 18919

This theorem is referenced by: fsuppssind 42616

W3C validator