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Theorem gsumval3eu 19505

Description: The group sum as defined in gsumval3a 19504 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 2948	. . . . 5 ⊢ (𝜑 → ¬ 𝑊 = ∅)
3		gsumval3a.t	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
4		fz1f1o 15422	. . . . . . 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 861	. . . . 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 496	. . 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 6787	. . . . . . 7 ⊢ (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∈ V
12		biidd 261	. . . . . . 7 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) → (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊))
13	11, 12	ceqsexv 3479	. . . . . 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 274	. . . . 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 274	. . 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 653	. . . . . 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 481	. . . . . . . . . 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 18393	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
25	24	3expb 1119	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
26	21, 25	sylan 580	. . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
29	28	sselda 3921	. . . . . . . . . . 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 714	. . . . . . . . . 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 770	. . . . . . . . . 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 18935	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
34	30, 31, 33	syl2anc 584	. . . . . . . . 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 18394	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
36	21, 35	sylan 580	. . . . . . . . 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 495	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ)
38	37	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (♯‘𝑊) ∈ ℕ)
39		nnuz 12621	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
40	38, 39	eleqtrdi 2849	. . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐹:𝐴⟶𝐵)
43	42	frnd 6608	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐹 ⊆ 𝐵)
44		simprr 770	. . . . . . . . . . 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 6728	. . . . . . . . . . 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 768	. . . . . . . . . 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 6739	. . . . . . . . . 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 584	. . . . . . . . 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 6716	. . . . . . . . . . . 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 6867	. . . . . . . . . . 11 ⊢ ((𝑔:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑔)‘𝑥) = (𝐹‘(𝑔‘𝑥)))
53	51, 52	sylan 580	. . . . . . . . . 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 6601	. . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑊 ⊆ 𝐴)
57	51, 56	fssd 6618	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝐴)
58	57	ffvelrnda 6961	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝑔‘𝑥) ∈ 𝐴)
59		fnfvelrn 6958	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐴) → (𝐹‘(𝑔‘𝑥)) ∈ ran 𝐹)
60	54, 58, 59	syl2an2r 682	. . . . . . . . . 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 2839	. . . . . . . . 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 6716	. . . . . . . . . . . . . . 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 6867	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((^◡𝑔 ∘ 𝑓)‘𝑘) = (^◡𝑔‘(𝑓‘𝑘)))
65	63, 64	sylan 580	. . . . . . . . . . . . 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 6778	. . . . . . . . . . . 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	ffvelrnda 6961	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓‘𝑘) ∈ 𝑊)
68		f1ocnvfv2 7149	. . . . . . . . . . . . 13 ⊢ ((𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ (𝑓‘𝑘) ∈ 𝑊) → (𝑔‘(^◡𝑔‘(𝑓‘𝑘))) = (𝑓‘𝑘))
69	44, 67, 68	syl2an2r 682	. . . . . . . . . . . 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 2779	. . . . . . . . . . 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 6778	. . . . . . . . . 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 6867	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝑊))⟶𝑊 ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) = (𝐹‘(𝑓‘𝑘)))
73	63, 72	sylan 580	. . . . . . . . . 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 6716	. . . . . . . . . . . . 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	ffvelrnda 6961	. . . . . . . . . . 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 6867	. . . . . . . . . . 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 682	. . . . . . . . . 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 2788	. . . . . . . . 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 13764	. . . . . . . 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 2755	. . . . . . . 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 246	. . . . . . 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 454	. . . . . 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 242	. . . . 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 242	. . 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 2742	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))
89	88	anbi2d 629	. . . . 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 6704	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ↔ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊))
92		coeq2 5767	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
93	92	seqeq3d 13729	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → seq1( + , (𝐹 ∘ 𝑓)) = seq1( + , (𝐹 ∘ 𝑔)))
94	93	fveq1d 6776	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))
95	94	eqeq2d 2749	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊))))
96	91, 95	anbi12d 631	. . . . 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 2617	. 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 583	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 396 ∨ wo 844 ∧ w3a 1086 ∀wal 1537 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∃!weu 2568 ≠ wne 2943 ⊆ wss 3887 ∅c0 4256 ^◡ccnv 5588 ran crn 5590 ∘ ccom 5593 Fn wfn 6428 ⟶wf 6429 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 1c1 10872 ℕcn 11973 ℤ_≥cuz 12582 ...cfz 13239 seqcseq 13721 ♯chash 14044 Basecbs 16912 +_gcplusg 16962 0_gc0g 17150 Mndcmnd 18385 Cntzccntz 18921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-cntz 18923

This theorem is referenced by: gsumval3lem2 19507

W3C validator