Theorem gsumval3eu 20151

Description: The group sum as defined in gsumval3a 20150 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.)

Hypotheses

Ref	Expression
gsumval3.b	⊢ 𝐵 = (Base‘𝐺)
gsumval3.0	⊢ 0 = (0g‘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))⟶1-1onto𝑊 ∧ 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 2964	. . . . 5 ⊢ (𝜑 → ¬ 𝑊 = ∅)
3		gsumval3a.t	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
4		fz1f1o 15908	. . . . . . 7 ⊢ (𝑊 ∈ Fin → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊)))
5	3, 4	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑊 = ∅ ∨ ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊)))
6	5	ord 878	. . . . 5 ⊢ (𝜑 → (¬ 𝑊 = ∅ → ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊)))
7	2, 6	mpd 16	. . . 4 ⊢ (𝜑 → ((♯‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊))
8	7	simprd 501	. . 3 ⊢ (𝜑 → ∃𝑓 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊)
9		excom 2202	. . . 4 ⊢ (∃𝑥∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑓∃𝑥(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))
10		exancom 1894	. . . . . 6 ⊢ (∃𝑥(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑥(𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊))
11		fvex 6924	. . . . . . 7 ⊢ (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∈ V
12		biidd 265	. . . . . . 7 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) → (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ↔ 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊))
13	11, 12	ceqsexv 3502	. . . . . 6 ⊢ (∃𝑥(𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊) ↔ 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊)
14	10, 13	bitri 278	. . . . 5 ⊢ (∃𝑥(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊)
15	14	exbii 1881	. . . 4 ⊢ (∃𝑓∃𝑥(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑓 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊)
16	9, 15	bitri 278	. . 3 ⊢ (∃𝑥∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑓 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊)
17	8, 16	sylibr 237	. 2 ⊢ (𝜑 → ∃𝑥∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))
18		exdistrv 1990	. . . 4 ⊢ (∃𝑓∃𝑔((𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) ↔ (∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))))
19		an4 669	. . . . . 6 ⊢ (((𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊) ∧ (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) ↔ ((𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))))
20		gsumval3.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Mnd)
21	20	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → 𝐺 ∈ Mnd)
22		gsumval3.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐺)
23		gsumval3.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝐺)
24	22, 23	mndcl 18963	. . . . . . . . . . 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))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ (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))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
29	28	sselda 3913	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑥 ∈ ran 𝐹) → 𝑥 ∈ (𝑍‘ran 𝐹))
30	29	adantrr 730	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹)) → 𝑥 ∈ (𝑍‘ran 𝐹))
31		simprr 785	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹)) → 𝑦 ∈ ran 𝐹)
32		gsumval3.z	. . . . . . . . . . 11 ⊢ 𝑍 = (Cntz‘𝐺)
33	23, 32	cntzi 19576	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
34	30, 31, 33	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ (𝑥 ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐹)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
35	22, 23	mndass 18964	. . . . . . . . . 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))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ (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))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → (♯‘𝑊) ∈ ℕ)
39		nnuz 13034	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
40	38, 39	eleqtrdi 2874	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → (♯‘𝑊) ∈ (ℤ≥‘1))
41		gsumval3.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
42	41	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → 𝐹:𝐴⟶𝐵)
43	42	frnd 6743	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → ran 𝐹 ⊆ 𝐵)
44		simprr 785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)
45		f1ocnv 6862	. . . . . . . . . . 11 ⊢ (𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 → ◡𝑔:𝑊⟶1-1onto(1...(♯‘𝑊)))
46	44, 45	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → ◡𝑔:𝑊⟶1-1onto(1...(♯‘𝑊)))
47		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊)
48		f1oco 6873	. . . . . . . . . 10 ⊢ ((◡𝑔:𝑊⟶1-1onto(1...(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊) → (◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))⟶1-1onto(1...(♯‘𝑊)))
49	46, 47, 48	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → (◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))⟶1-1onto(1...(♯‘𝑊)))
50		f1of 6849	. . . . . . . . . . . 12 ⊢ (𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 → 𝑔:(1...(♯‘𝑊))⟶𝑊)
51	44, 50	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝑊)
52		fvco3 7011	. . . . . . . . . . 11 ⊢ ((𝑔:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑔)‘𝑥) = (𝐹‘(𝑔‘𝑥)))
53	51, 52	sylan 592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑔)‘𝑥) = (𝐹‘(𝑔‘𝑥)))
54	42	ffnd 6735	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → 𝐹 Fn 𝐴)
55		gsumval3a.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊 ⊆ 𝐴)
56	55	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → 𝑊 ⊆ 𝐴)
57	51, 56	fssd 6752	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝐴)
58	57	ffvelcdmda 7111	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝑔‘𝑥) ∈ 𝐴)
59		fnfvelrn 7107	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐴) → (𝐹‘(𝑔‘𝑥)) ∈ ran 𝐹)
60	54, 58, 59	syl2an2r 698	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑔‘𝑥)) ∈ ran 𝐹)
61	53, 60	eqeltrd 2864	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑔)‘𝑥) ∈ ran 𝐹)
62		f1of 6849	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 → 𝑓:(1...(♯‘𝑊))⟶𝑊)
63	47, 62	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → 𝑓:(1...(♯‘𝑊))⟶𝑊)
64		fvco3 7011	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((◡𝑔 ∘ 𝑓)‘𝑘) = (◡𝑔‘(𝑓‘𝑘)))
65	63, 64	sylan 592	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((◡𝑔 ∘ 𝑓)‘𝑘) = (◡𝑔‘(𝑓‘𝑘)))
66	65	fveq2d 6915	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘((◡𝑔 ∘ 𝑓)‘𝑘)) = (𝑔‘(◡𝑔‘(𝑓‘𝑘))))
67	63	ffvelcdmda 7111	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓‘𝑘) ∈ 𝑊)
68		f1ocnvfv2 7311	. . . . . . . . . . . . 13 ⊢ ((𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ (𝑓‘𝑘) ∈ 𝑊) → (𝑔‘(◡𝑔‘(𝑓‘𝑘))) = (𝑓‘𝑘))
69	44, 67, 68	syl2an2r 698	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘(◡𝑔‘(𝑓‘𝑘))) = (𝑓‘𝑘))
70	66, 69	eqtr2d 2799	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓‘𝑘) = (𝑔‘((◡𝑔 ∘ 𝑓)‘𝑘)))
71	70	fveq2d 6915	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑓‘𝑘)) = (𝐹‘(𝑔‘((◡𝑔 ∘ 𝑓)‘𝑘))))
72		fvco3 7011	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) = (𝐹‘(𝑓‘𝑘)))
73	63, 72	sylan 592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) = (𝐹‘(𝑓‘𝑘)))
74		f1of 6849	. . . . . . . . . . . . 13 ⊢ ((◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))⟶1-1onto(1...(♯‘𝑊)) → (◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))⟶(1...(♯‘𝑊)))
75	49, 74	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → (◡𝑔 ∘ 𝑓):(1...(♯‘𝑊))⟶(1...(♯‘𝑊)))
76	75	ffvelcdmda 7111	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((◡𝑔 ∘ 𝑓)‘𝑘) ∈ (1...(♯‘𝑊)))
77		fvco3 7011	. . . . . . . . . . 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))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑔)‘((◡𝑔 ∘ 𝑓)‘𝑘)) = (𝐹‘(𝑔‘((◡𝑔 ∘ 𝑓)‘𝑘))))
79	71, 73, 78	3eqtr4d 2808	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) = ((𝐹 ∘ 𝑔)‘((◡𝑔 ∘ 𝑓)‘𝑘)))
80	26, 34, 36, 40, 43, 49, 61, 79	seqf1o 14216	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))
81		eqeq12 2780	. . . . . . . 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))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊)) → ((𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊))) → 𝑥 = 𝑦))
83	82	expimpd 459	. . . . . 6 ⊢ (𝜑 → (((𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊) ∧ (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
84	19, 83	biimtrrid 246	. . . . 5 ⊢ (𝜑 → (((𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
85	84	exlimdvv 1969	. . . 4 ⊢ (𝜑 → (∃𝑓∃𝑔((𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
86	18, 85	biimtrrid 246	. . 3 ⊢ (𝜑 → ((∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
87	86	alrimivv 1963	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦((∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
88		eqeq1 2767	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))
89	88	anbi2d 642	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))
90	89	exbidv 1956	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))
91		f1oeq1 6837	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ↔ 𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊))
92		coeq2 5843	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
93	92	seqeq3d 14182	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → seq1( + , (𝐹 ∘ 𝑓)) = seq1( + , (𝐹 ∘ 𝑔)))
94	93	fveq1d 6913	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))
95	94	eqeq2d 2774	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊))))
96	91, 95	anbi12d 644	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ (𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))))
97	96	cbvexvw 2072	. . . 4 ⊢ (∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑔(𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊))))
98	90, 97	bitrdi 290	. . 3 ⊢ (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ ∃𝑔(𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))))
99	98	eu4 2643	. 2 ⊢ (∃!𝑥∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ (∃𝑥∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ ∀𝑥∀𝑦((∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦)))
100	17, 87, 99	sylanbrc 595	1 ⊢ (𝜑 → ∃!𝑥∃𝑓(𝑓:(1...(♯‘𝑊))⟶1-1onto𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃!weu 2596   ≠ wne 2959   ⊆ wss 3900  ∅c0 4281  ^◡ccnv 5655  ran crn 5657   ∘ ccom 5660   Fn wfn 6556  ⟶wf 6557  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446  Fincfn 9004  1c1 11231  ℕcn 12365  ℤ_≥cuz 12995  ...cfz 13669  seqcseq 14174  ♯chash 14504  Basecbs 17419  +_gcplusg 17460  0_gc0g 17642  Mndcmnd 18955  Cntzccntz 19562

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-er 8748  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-card 10050  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-n0 12637  df-z 12724  df-uz 12996  df-fz 13670  df-fzo 13819  df-seq 14175  df-hash 14505  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-cntz 19564

This theorem is used by:  gsumval3lem2  20153