Theorem gsumf1ofi 14137

Description: Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.)

Hypotheses

Ref	Expression
gsumclfi.b	⊢ 𝐵 = (Base‘𝐺)
gsumclfi.z	⊢ 0 = (0g‘𝐺)
gsumclfi.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsumclfi.a	⊢ (𝜑 → 𝐴 ∈ Fin)
gsumclfi.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumf1o.h	⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)

Assertion

Ref	Expression
gsumf1ofi	⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))

Proof of Theorem gsumf1ofi

Dummy variables 𝑥 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumclfi.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		isfinite4im 11209	. . . 4 ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴)
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → (1...(♯‘𝐴)) ≈ 𝐴)
4		bren 7020	. . 3 ⊢ ((1...(♯‘𝐴)) ≈ 𝐴 ↔ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
5	3, 4	sylib 122	. 2 ⊢ (𝜑 → ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
6		gsumf1o.h	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)
7		f1ocnv 5647	. . . . . . . . . 10 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → ◡𝐻:𝐴–1-1-onto→𝐶)
8	6, 7	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ◡𝐻:𝐴–1-1-onto→𝐶)
9		f1oeng 7033	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ ◡𝐻:𝐴–1-1-onto→𝐶) → 𝐴 ≈ 𝐶)
10	1, 8, 9	syl2anc 415	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≈ 𝐶)
11	10	ensymd 7060	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≈ 𝐴)
12		enfii 7166	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ≈ 𝐴) → 𝐶 ∈ Fin)
13	1, 11, 12	syl2anc 415	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Fin)
14		isfinite4im 11209	. . . . . 6 ⊢ (𝐶 ∈ Fin → (1...(♯‘𝐶)) ≈ 𝐶)
15	13, 14	syl 14	. . . . 5 ⊢ (𝜑 → (1...(♯‘𝐶)) ≈ 𝐶)
16		bren 7020	. . . . 5 ⊢ ((1...(♯‘𝐶)) ≈ 𝐶 ↔ ∃𝑔 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶)
17	15, 16	sylib 122	. . . 4 ⊢ (𝜑 → ∃𝑔 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶)
18	17	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∃𝑔 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶)
19		gsumclfi.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
20		gsumclfi.z	. . . . . 6 ⊢ 0 = (0g‘𝐺)
21		gsumclfi.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ CMnd)
22	21	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝐺 ∈ CMnd)
23		1zzd 9650	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 1 ∈ ℤ)
24	1	ad2antrr 492	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝐴 ∈ Fin)
25		hashcl 11198	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
26	24, 25	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (♯‘𝐴) ∈ ℕ0)
27	26	nn0zd 9745	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (♯‘𝐴) ∈ ℤ)
28		gsumclfi.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
29	28	ad2antrr 492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝐹:𝐴⟶𝐵)
30		f1of 5634	. . . . . . . 8 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
31	30	ad2antlr 493	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
32	29, 31	fcod 5548	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐹 ∘ 𝑓):(1...(♯‘𝐴))⟶𝐵)
33		f1ocnv 5647	. . . . . . . 8 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(1...(♯‘𝐴)))
34	33	ad2antlr 493	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → ◡𝑓:𝐴–1-1-onto→(1...(♯‘𝐴)))
35	6	ad2antrr 492	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝐻:𝐶–1-1-onto→𝐴)
36		simpr 110	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶)
37	10	ad2antrr 492	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝐴 ≈ 𝐶)
38	13	ad2antrr 492	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝐶 ∈ Fin)
39		hashen 11201	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐶) ↔ 𝐴 ≈ 𝐶))
40	24, 38, 39	syl2anc 415	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → ((♯‘𝐴) = (♯‘𝐶) ↔ 𝐴 ≈ 𝐶))
41	37, 40	mpbird 167	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (♯‘𝐴) = (♯‘𝐶))
42	41	oveq2d 6091	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (1...(♯‘𝐴)) = (1...(♯‘𝐶)))
43	42	f1oeq2d 5630	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝑔:(1...(♯‘𝐴))–1-1-onto→𝐶 ↔ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶))
44	36, 43	mpbird 167	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐶)
45		f1oco 5657	. . . . . . . 8 ⊢ ((𝐻:𝐶–1-1-onto→𝐴 ∧ 𝑔:(1...(♯‘𝐴))–1-1-onto→𝐶) → (𝐻 ∘ 𝑔):(1...(♯‘𝐴))–1-1-onto→𝐴)
46	35, 44, 45	syl2anc 415	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐻 ∘ 𝑔):(1...(♯‘𝐴))–1-1-onto→𝐴)
47		f1oco 5657	. . . . . . 7 ⊢ ((◡𝑓:𝐴–1-1-onto→(1...(♯‘𝐴)) ∧ (𝐻 ∘ 𝑔):(1...(♯‘𝐴))–1-1-onto→𝐴) → (◡𝑓 ∘ (𝐻 ∘ 𝑔)):(1...(♯‘𝐴))–1-1-onto→(1...(♯‘𝐴)))
48	34, 46, 47	syl2anc 415	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (◡𝑓 ∘ (𝐻 ∘ 𝑔)):(1...(♯‘𝐴))–1-1-onto→(1...(♯‘𝐴)))
49	19, 20, 22, 23, 27, 32, 48	gzsumreidx 14118	. . . . 5 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐺 Σgz (𝐹 ∘ 𝑓)) = (𝐺 Σgz ((𝐹 ∘ 𝑓) ∘ (◡𝑓 ∘ (𝐻 ∘ 𝑔)))))
50		coass 5301	. . . . . . . 8 ⊢ (((𝐹 ∘ 𝑓) ∘ ◡𝑓) ∘ (𝐻 ∘ 𝑔)) = ((𝐹 ∘ 𝑓) ∘ (◡𝑓 ∘ (𝐻 ∘ 𝑔)))
51		f1of 5634	. . . . . . . . . . . . 13 ⊢ (◡𝑓:𝐴–1-1-onto→(1...(♯‘𝐴)) → ◡𝑓:𝐴⟶(1...(♯‘𝐴)))
52	34, 51	syl 14	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → ◡𝑓:𝐴⟶(1...(♯‘𝐴)))
53	32, 52	fcod 5548	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → ((𝐹 ∘ 𝑓) ∘ ◡𝑓):𝐴⟶𝐵)
54	53	ffnd 5529	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → ((𝐹 ∘ 𝑓) ∘ ◡𝑓) Fn 𝐴)
55	29	ffnd 5529	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝐹 Fn 𝐴)
56		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
57	56	ad2antrr 492	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) ∧ 𝑥 ∈ 𝐴) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
58	57, 33, 51	3syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) ∧ 𝑥 ∈ 𝐴) → ◡𝑓:𝐴⟶(1...(♯‘𝐴)))
59		fvco3 5770	. . . . . . . . . . . 12 ⊢ ((◡𝑓:𝐴⟶(1...(♯‘𝐴)) ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∘ 𝑓) ∘ ◡𝑓)‘𝑥) = ((𝐹 ∘ 𝑓)‘(◡𝑓‘𝑥)))
60	58, 59	sylancom 424	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∘ 𝑓) ∘ ◡𝑓)‘𝑥) = ((𝐹 ∘ 𝑓)‘(◡𝑓‘𝑥)))
61	57, 30	syl 14	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) ∧ 𝑥 ∈ 𝐴) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
62	52	ffvelcdmda 5834	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) ∧ 𝑥 ∈ 𝐴) → (◡𝑓‘𝑥) ∈ (1...(♯‘𝐴)))
63		fvco3 5770	. . . . . . . . . . . 12 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ (◡𝑓‘𝑥) ∈ (1...(♯‘𝐴))) → ((𝐹 ∘ 𝑓)‘(◡𝑓‘𝑥)) = (𝐹‘(𝑓‘(◡𝑓‘𝑥))))
64	61, 62, 63	syl2anc 415	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘ 𝑓)‘(◡𝑓‘𝑥)) = (𝐹‘(𝑓‘(◡𝑓‘𝑥))))
65		f1ocnvfv2 5974	. . . . . . . . . . . . 13 ⊢ ((𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓‘(◡𝑓‘𝑥)) = 𝑥)
66	57, 65	sylancom 424	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) ∧ 𝑥 ∈ 𝐴) → (𝑓‘(◡𝑓‘𝑥)) = 𝑥)
67	66	fveq2d 5694	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) ∧ 𝑥 ∈ 𝐴) → (𝐹‘(𝑓‘(◡𝑓‘𝑥))) = (𝐹‘𝑥))
68	60, 64, 67	3eqtrd 2275	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) ∧ 𝑥 ∈ 𝐴) → (((𝐹 ∘ 𝑓) ∘ ◡𝑓)‘𝑥) = (𝐹‘𝑥))
69	54, 55, 68	eqfnfvd 5800	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → ((𝐹 ∘ 𝑓) ∘ ◡𝑓) = 𝐹)
70	69	coeq1d 4936	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (((𝐹 ∘ 𝑓) ∘ ◡𝑓) ∘ (𝐻 ∘ 𝑔)) = (𝐹 ∘ (𝐻 ∘ 𝑔)))
71	50, 70	eqtr3id 2285	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → ((𝐹 ∘ 𝑓) ∘ (◡𝑓 ∘ (𝐻 ∘ 𝑔))) = (𝐹 ∘ (𝐻 ∘ 𝑔)))
72		coass 5301	. . . . . . 7 ⊢ ((𝐹 ∘ 𝐻) ∘ 𝑔) = (𝐹 ∘ (𝐻 ∘ 𝑔))
73	71, 72	eqtr4di 2289	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → ((𝐹 ∘ 𝑓) ∘ (◡𝑓 ∘ (𝐻 ∘ 𝑔))) = ((𝐹 ∘ 𝐻) ∘ 𝑔))
74	73	oveq2d 6091	. . . . 5 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐺 Σgz ((𝐹 ∘ 𝑓) ∘ (◡𝑓 ∘ (𝐻 ∘ 𝑔)))) = (𝐺 Σgz ((𝐹 ∘ 𝐻) ∘ 𝑔)))
75	49, 74	eqtrd 2271	. . . 4 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐺 Σgz (𝐹 ∘ 𝑓)) = (𝐺 Σgz ((𝐹 ∘ 𝐻) ∘ 𝑔)))
76	21	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐺 ∈ CMnd)
77	28	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐹:𝐴⟶𝐵)
78	1	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝐴 ∈ Fin)
79	19, 76, 77, 78, 56	gsumvalfi 14129	. . . . 5 ⊢ ((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝐺 Σg 𝐹) = (𝐺 Σgz (𝐹 ∘ 𝑓)))
80	79	adantr 276	. . . 4 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐺 Σg 𝐹) = (𝐺 Σgz (𝐹 ∘ 𝑓)))
81		f1of 5634	. . . . . . . 8 ⊢ (𝐻:𝐶–1-1-onto→𝐴 → 𝐻:𝐶⟶𝐴)
82	6, 81	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝐶⟶𝐴)
83	82	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝐻:𝐶⟶𝐴)
84	29, 83	fcod 5548	. . . . 5 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐹 ∘ 𝐻):𝐶⟶𝐵)
85	19, 22, 84, 38, 36	gsumvalfi 14129	. . . 4 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐺 Σg (𝐹 ∘ 𝐻)) = (𝐺 Σgz ((𝐹 ∘ 𝐻) ∘ 𝑔)))
86	75, 80, 85	3eqtr4d 2281	. . 3 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑔:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
87	18, 86	exlimddv 1954	. 2 ⊢ ((𝜑 ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
88	5, 87	exlimddv 1954	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209   class class class wbr 4125  ^◡ccnv 4768   ∘ ccom 4773  ⟶wf 5368  –1-1-onto→wf1o 5371  ‘cfv 5372  (class class class)co 6075   ≈ cen 7010  Fincfn 7012  1c1 8170  ℕ₀cn0 9542  ...cfz 10390  ♯chash 11192  Basecbs 13330  0_gc0g 13587   Σ_gz cgzsu 13588  CMndccmn 14064   Σ_g cgsu 14127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-ihash 11193  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-gzsum 13590  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-cmn 14066  df-gsumfi 14128

This theorem is referenced by:  lgseisenlem3  16105