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Theorem gsumval3eu 20098

Description: The group sum as defined in gsumval3a 20097 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 2961	. . . . 5 ⊢ (𝜑 → ¬ 𝑊 = ∅)
3		gsumval3a.t	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
4		fz1f1o 15856	. . . . . . 7 ⊢ (𝑊 ∈ Fin → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)))
5	3, 4	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)))
6	5	ord 878	. . . . 5 ⊢ (𝜑 → (¬ 𝑊 = ∅ → ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)))
7	2, 6	mpd 16	. . . 4 ⊢ (𝜑 → ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊))
8	7	simprd 501	. . 3 ⊢ (𝜑 → ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)
9		excom 2199	. . . 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 1894	. . . . . 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 6890	. . . . . . 7 ⊢ (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∈ V
12		biidd 265	. . . . . . 7 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) → (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊))
13	11, 12	ceqsexv 3499	. . . . . 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 278	. . . . 5 ⊢ (∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)
15	14	exbii 1881	. . . 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 278	. . 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 237	. 2 ⊢ (𝜑 → ∃𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))
18		exdistrv 1988	. . . 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 669	. . . . . 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 486	. . . . . . . . . 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 18911	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
25	24	3expb 1138	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
26	21, 25	sylan 592	. . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
29	28	sselda 3931	. . . . . . . . . . 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 730	. . . . . . . . . 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 785	. . . . . . . . . 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 19523	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
34	30, 31, 33	syl2anc 596	. . . . . . . . 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 18912	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
36	21, 35	sylan 592	. . . . . . . . 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 500	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ)
38	37	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (♯‘𝑊) ∈ ℕ)
39		nnuz 12985	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
40	38, 39	eleqtrdi 2871	. . . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐹:𝐴⟶𝐵)
43	42	frnd 6710	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐹 ⊆ 𝐵)
44		simprr 785	. . . . . . . . . . 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 6829	. . . . . . . . . . 11 ⊢ (𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 → ^◡𝑔:𝑊–1-1-onto→(1...(♯‘𝑊)))
46	44, 45	syl 18	. . . . . . . . . 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 783	. . . . . . . . . 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 6840	. . . . . . . . . 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 596	. . . . . . . . 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 6816	. . . . . . . . . . . 12 ⊢ (𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 → 𝑔:(1...(♯‘𝑊))⟶𝑊)
51	44, 50	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝑊)
52		fvco3 6977	. . . . . . . . . . 11 ⊢ ((𝑔:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑔)‘𝑥) = (𝐹‘(𝑔‘𝑥)))
53	51, 52	sylan 592	. . . . . . . . . 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 6702	. . . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑊 ⊆ 𝐴)
57	51, 56	fssd 6719	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝐴)
58	57	ffvelcdmda 7076	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝑔‘𝑥) ∈ 𝐴)
59		fnfvelrn 7072	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐴) → (𝐹‘(𝑔‘𝑥)) ∈ ran 𝐹)
60	54, 58, 59	syl2an2r 698	. . . . . . . . . 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 2861	. . . . . . . . 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 6816	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 → 𝑓:(1...(♯‘𝑊))⟶𝑊)
63	47, 62	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑓:(1...(♯‘𝑊))⟶𝑊)
64		fvco3 6977	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((^◡𝑔 ∘ 𝑓)‘𝑘) = (^◡𝑔‘(𝑓‘𝑘)))
65	63, 64	sylan 592	. . . . . . . . . . . . 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 6881	. . . . . . . . . . . 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 7076	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓‘𝑘) ∈ 𝑊)
68		f1ocnvfv2 7277	. . . . . . . . . . . . 13 ⊢ ((𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ (𝑓‘𝑘) ∈ 𝑊) → (𝑔‘(^◡𝑔‘(𝑓‘𝑘))) = (𝑓‘𝑘))
69	44, 67, 68	syl2an2r 698	. . . . . . . . . . . 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 2797	. . . . . . . . . . 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 6881	. . . . . . . . . 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 6977	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) = (𝐹‘(𝑓‘𝑘)))
73	63, 72	sylan 592	. . . . . . . . . 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 6816	. . . . . . . . . . . . 13 ⊢ ((^◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)) → (^◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))⟶(1...(♯‘𝑊)))
75	49, 74	syl 18	. . . . . . . . . . . 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 7076	. . . . . . . . . . 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 6977	. . . . . . . . . . 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 698	. . . . . . . . . 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 2806	. . . . . . . . 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 14166	. . . . . . . 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 2778	. . . . . . . 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 250	. . . . . . 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 459	. . . . . 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	biimtrrid 246	. . . . 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 1967	. . . 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	biimtrrid 246	. . 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 1961	. 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 2765	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))
89	88	anbi2d 642	. . . . 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 1954	. . . 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 6804	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ↔ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊))
92		coeq2 5836	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
93	92	seqeq3d 14132	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → seq1( + , (𝐹 ∘ 𝑓)) = seq1( + , (𝐹 ∘ 𝑔)))
94	93	fveq1d 6879	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))
95	94	eqeq2d 2772	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊))))
96	91, 95	anbi12d 644	. . . . 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 2070	. . . 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 290	. . 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 2641	. 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 595	1 ⊢ (𝜑 → ∃!𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 ∀wal 1568 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ∃!weu 2594 ≠ wne 2956 ⊆ wss 3899 ∅c0 4279 ^◡ccnv 5650 ran crn 5652 ∘ ccom 5655 Fn wfn 6526 ⟶wf 6527 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7412 Fincfn 8957 1c1 11182 ℕcn 12316 ℤ_≥cuz 12946 ...cfz 13620 seqcseq 14124 ♯chash 14454 Basecbs 17367 +_gcplusg 17408 0_gc0g 17590 Mndcmnd 18903 Cntzccntz 19509

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-er 8701 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-n0 12588 df-z 12675 df-uz 12947 df-fz 13621 df-fzo 13769 df-seq 14125 df-hash 14455 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-cntz 19511

This theorem is used by: gsumval3lem2 20100

W3C validator