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Theorem gsumval3eu 19599

Description: The group sum as defined in gsumval3a 19598 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.)

**Hypotheses**
Ref	Expression
gsumval3.b	⊢ 𝐵 = (Base‘𝐺)
gsumval3.0	⊢ 0 = (0_g‘𝐺)
gsumval3.p	⊢ + = (+_g‘𝐺)
gsumval3.z	⊢ 𝑍 = (Cntz‘𝐺)
gsumval3.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
gsumval3.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumval3.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumval3.c	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumval3a.t	⊢ (𝜑 → 𝑊 ∈ Fin)
gsumval3a.n	⊢ (𝜑 → 𝑊 ≠ ∅)
gsumval3a.s	⊢ (𝜑 → 𝑊 ⊆ 𝐴)

**Assertion**
Ref	Expression
gsumval3eu	⊢ (𝜑 → ∃!𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))

Distinct variable groups: 𝑥,𝑓, + 𝐴,𝑓,𝑥 𝜑,𝑓,𝑥 𝑥, 0 𝑓,𝐺,𝑥 𝑥,𝑉 𝐵,𝑓,𝑥 𝑓,𝐹,𝑥 𝑓,𝑊,𝑥 Allowed substitution hints: 𝑉(𝑓) 0 (𝑓) 𝑍(𝑥,𝑓)

Proof of Theorem gsumval3eu Dummy variables 𝑔 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumval3a.n	. . . . . 6 ⊢ (𝜑 → 𝑊 ≠ ∅)
2	1	neneqd 2946	. . . . 5 ⊢ (𝜑 → ¬ 𝑊 = ∅)
3		gsumval3a.t	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
4		fz1f1o 15521	. . . . . . 7 ⊢ (𝑊 ∈ Fin → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)))
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)))
6	5	ord 862	. . . . 5 ⊢ (𝜑 → (¬ 𝑊 = ∅ → ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)))
7	2, 6	mpd 15	. . . 4 ⊢ (𝜑 → ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊))
8	7	simprd 497	. . 3 ⊢ (𝜑 → ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)
9		excom 2162	. . . 4 ⊢ (∃𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑓∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))
10		exancom 1864	. . . . . 6 ⊢ (∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑥(𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊))
11		fvex 6842	. . . . . . 7 ⊢ (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∈ V
12		biidd 262	. . . . . . 7 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) → (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊))
13	11, 12	ceqsexv 3490	. . . . . 6 ⊢ (∃𝑥(𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊) ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)
14	10, 13	bitri 275	. . . . 5 ⊢ (∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)
15	14	exbii 1850	. . . 4 ⊢ (∃𝑓∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)
16	9, 15	bitri 275	. . 3 ⊢ (∃𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)
17	8, 16	sylibr 233	. 2 ⊢ (𝜑 → ∃𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))
18		exdistrv 1959	. . . 4 ⊢ (∃𝑓∃𝑔((𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) ↔ (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))))
19		an4 654	. . . . . 6 ⊢ (((𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊) ∧ (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) ↔ ((𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))))
20		gsumval3.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Mnd)
21	20	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐺 ∈ Mnd)
22		gsumval3.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
23		gsumval3.p	. . . . . . . . . . . 12 ⊢ + = (+_g‘𝐺)
24	22, 23	mndcl 18490	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
25	24	3expb 1120	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
26	21, 25	sylan 581	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
27		gsumval3.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
28	27	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
29	28	sselda 3935	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ ran 𝐹) → 𝑥 ∈ (𝑍‘ran 𝐹))
30	29	adantrr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹)) → 𝑥 ∈ (𝑍‘ran 𝐹))
31		simprr 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹)) → 𝑦 ∈ ran 𝐹)
32		gsumval3.z	. . . . . . . . . . 11 ⊢ 𝑍 = (Cntz‘𝐺)
33	23, 32	cntzi 19031	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
34	30, 31, 33	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
35	22, 23	mndass 18491	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
36	21, 35	sylan 581	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
37	7	simpld 496	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ)
38	37	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (♯‘𝑊) ∈ ℕ)
39		nnuz 12726	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
40	38, 39	eleqtrdi 2848	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (♯‘𝑊) ∈ (ℤ_≥‘1))
41		gsumval3.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
42	41	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐹:𝐴⟶𝐵)
43	42	frnd 6663	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐹 ⊆ 𝐵)
44		simprr 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)
45		f1ocnv 6783	. . . . . . . . . . 11 ⊢ (𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 → ^◡𝑔:𝑊–1-1-onto→(1...(♯‘𝑊)))
46	44, 45	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ^◡𝑔:𝑊–1-1-onto→(1...(♯‘𝑊)))
47		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)
48		f1oco 6794	. . . . . . . . . 10 ⊢ ((^◡𝑔:𝑊–1-1-onto→(1...(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊) → (^◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)))
49	46, 47, 48	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (^◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)))
50		f1of 6771	. . . . . . . . . . . 12 ⊢ (𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 → 𝑔:(1...(♯‘𝑊))⟶𝑊)
51	44, 50	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝑊)
52		fvco3 6927	. . . . . . . . . . 11 ⊢ ((𝑔:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑔)‘𝑥) = (𝐹‘(𝑔‘𝑥)))
53	51, 52	sylan 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑔)‘𝑥) = (𝐹‘(𝑔‘𝑥)))
54	42	ffnd 6656	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐹 Fn 𝐴)
55		gsumval3a.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊 ⊆ 𝐴)
56	55	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑊 ⊆ 𝐴)
57	51, 56	fssd 6673	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝐴)
58	57	ffvelcdmda 7021	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝑔‘𝑥) ∈ 𝐴)
59		fnfvelrn 7018	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐴) → (𝐹‘(𝑔‘𝑥)) ∈ ran 𝐹)
60	54, 58, 59	syl2an2r 683	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑔‘𝑥)) ∈ ran 𝐹)
61	53, 60	eqeltrd 2838	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑔)‘𝑥) ∈ ran 𝐹)
62		f1of 6771	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → 𝑓:(1...(♯‘𝑊))⟶𝑊)
63	47, 62	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑓:(1...(♯‘𝑊))⟶𝑊)
64		fvco3 6927	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((^◡𝑔 ∘ 𝑓)‘𝑘) = (^◡𝑔‘(𝑓‘𝑘)))
65	63, 64	sylan 581	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((^◡𝑔 ∘ 𝑓)‘𝑘) = (^◡𝑔‘(𝑓‘𝑘)))
66	65	fveq2d 6833	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘((^◡𝑔 ∘ 𝑓)‘𝑘)) = (𝑔‘(^◡𝑔‘(𝑓‘𝑘))))
67	63	ffvelcdmda 7021	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓‘𝑘) ∈ 𝑊)
68		f1ocnvfv2 7209	. . . . . . . . . . . . 13 ⊢ ((𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ (𝑓‘𝑘) ∈ 𝑊) → (𝑔‘(^◡𝑔‘(𝑓‘𝑘))) = (𝑓‘𝑘))
69	44, 67, 68	syl2an2r 683	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘(^◡𝑔‘(𝑓‘𝑘))) = (𝑓‘𝑘))
70	66, 69	eqtr2d 2778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓‘𝑘) = (𝑔‘((^◡𝑔 ∘ 𝑓)‘𝑘)))
71	70	fveq2d 6833	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑓‘𝑘)) = (𝐹‘(𝑔‘((^◡𝑔 ∘ 𝑓)‘𝑘))))
72		fvco3 6927	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) = (𝐹‘(𝑓‘𝑘)))
73	63, 72	sylan 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) = (𝐹‘(𝑓‘𝑘)))
74		f1of 6771	. . . . . . . . . . . . 13 ⊢ ((^◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)) → (^◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))⟶(1...(♯‘𝑊)))
75	49, 74	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (^◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))⟶(1...(♯‘𝑊)))
76	75	ffvelcdmda 7021	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((^◡𝑔 ∘ 𝑓)‘𝑘) ∈ (1...(♯‘𝑊)))
77		fvco3 6927	. . . . . . . . . . 11 ⊢ ((𝑔:(1...(♯‘𝑊))⟶𝐴 ∧ ((^◡𝑔 ∘ 𝑓)‘𝑘) ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑔)‘((^◡𝑔 ∘ 𝑓)‘𝑘)) = (𝐹‘(𝑔‘((^◡𝑔 ∘ 𝑓)‘𝑘))))
78	57, 76, 77	syl2an2r 683	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑔)‘((^◡𝑔 ∘ 𝑓)‘𝑘)) = (𝐹‘(𝑔‘((^◡𝑔 ∘ 𝑓)‘𝑘))))
79	71, 73, 78	3eqtr4d 2787	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) = ((𝐹 ∘ 𝑔)‘((^◡𝑔 ∘ 𝑓)‘𝑘)))
80	26, 34, 36, 40, 43, 49, 61, 79	seqf1o 13869	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))
81		eqeq12 2754	. . . . . . . 8 ⊢ ((𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊))) → (𝑥 = 𝑦 ↔ (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊))))
82	80, 81	syl5ibrcom 247	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ((𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊))) → 𝑥 = 𝑦))
83	82	expimpd 455	. . . . . 6 ⊢ (𝜑 → (((𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊) ∧ (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
84	19, 83	syl5bir 243	. . . . 5 ⊢ (𝜑 → (((𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
85	84	exlimdvv 1937	. . . 4 ⊢ (𝜑 → (∃𝑓∃𝑔((𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
86	18, 85	syl5bir 243	. . 3 ⊢ (𝜑 → ((∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
87	86	alrimivv 1931	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦((∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
88		eqeq1 2741	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))
89	88	anbi2d 630	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))
90	89	exbidv 1924	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))
91		f1oeq1 6759	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ↔ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊))
92		coeq2 5804	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
93	92	seqeq3d 13834	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → seq1( + , (𝐹 ∘ 𝑓)) = seq1( + , (𝐹 ∘ 𝑔)))
94	93	fveq1d 6831	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))
95	94	eqeq2d 2748	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊))))
96	91, 95	anbi12d 632	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ (𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))))
97	96	cbvexvw 2040	. . . 4 ⊢ (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊))))
98	90, 97	bitrdi 287	. . 3 ⊢ (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))))
99	98	eu4 2616	. 2 ⊢ (∃!𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ (∃𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ ∀𝑥∀𝑦((∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦)))
100	17, 87, 99	sylanbrc 584	1 ⊢ (𝜑 → ∃!𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 845 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∃!weu 2567 ≠ wne 2941 ⊆ wss 3901 ∅c0 4273 ^◡ccnv 5623 ran crn 5625 ∘ ccom 5628 Fn wfn 6478 ⟶wf 6479 –1-1-onto→wf1o 6482 ‘cfv 6483 (class class class)co 7341 Fincfn 8808 1c1 10977 ℕcn 12078 ℤ_≥cuz 12687 ...cfz 13344 seqcseq 13826 ♯chash 14149 Basecbs 17009 +_gcplusg 17059 0_gc0g 17247 Mndcmnd 18482 Cntzccntz 19017

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5233 ax-sep 5247 ax-nul 5254 ax-pow 5312 ax-pr 5376 ax-un 7654 ax-cnex 11032 ax-resscn 11033 ax-1cn 11034 ax-icn 11035 ax-addcl 11036 ax-addrcl 11037 ax-mulcl 11038 ax-mulrcl 11039 ax-mulcom 11040 ax-addass 11041 ax-mulass 11042 ax-distr 11043 ax-i2m1 11044 ax-1ne0 11045 ax-1rid 11046 ax-rnegex 11047 ax-rrecex 11048 ax-cnre 11049 ax-pre-lttri 11050 ax-pre-lttrn 11051 ax-pre-ltadd 11052 ax-pre-mulgt0 11053

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3731 df-csb 3847 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3920 df-nul 4274 df-if 4478 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4857 df-int 4899 df-iun 4947 df-br 5097 df-opab 5159 df-mpt 5180 df-tr 5214 df-id 5522 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5579 df-we 5581 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6242 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6435 df-fun 6485 df-fn 6486 df-f 6487 df-f1 6488 df-fo 6489 df-f1o 6490 df-fv 6491 df-riota 7297 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7785 df-1st 7903 df-2nd 7904 df-frecs 8171 df-wrecs 8202 df-recs 8276 df-rdg 8315 df-1o 8371 df-er 8573 df-en 8809 df-dom 8810 df-sdom 8811 df-fin 8812 df-card 9800 df-pnf 11116 df-mnf 11117 df-xr 11118 df-ltxr 11119 df-le 11120 df-sub 11312 df-neg 11313 df-nn 12079 df-n0 12339 df-z 12425 df-uz 12688 df-fz 13345 df-fzo 13488 df-seq 13827 df-hash 14150 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-cntz 19019

This theorem is referenced by: gsumval3lem2 19601

W3C validator