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Theorem gsumval3eu 18305

Description: The group sum as defined in gsumval3a 18304 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 2799	. . . . 5 ⊢ (𝜑 → ¬ 𝑊 = ∅)
3		gsumval3a.t	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Fin)
4		fz1f1o 14441	. . . . . . 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 392	. . . . 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 479	. . 3 ⊢ (𝜑 → ∃𝑓 𝑓:(1...(#‘𝑊))–1-1-onto→𝑊)
9		excom 2042	. . . 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 1787	. . . . . 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 6201	. . . . . . 7 ⊢ (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊)) ∈ V
12		biidd 252	. . . . . . 7 ⊢ (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊)) → (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ↔ 𝑓:(1...(#‘𝑊))–1-1-onto→𝑊))
13	11, 12	ceqsexv 3242	. . . . . 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 264	. . . . 5 ⊢ (∃𝑥(𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊))) ↔ 𝑓:(1...(#‘𝑊))–1-1-onto→𝑊)
15	14	exbii 1774	. . . 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 264	. . 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 224	. 2 ⊢ (𝜑 → ∃𝑥∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊))))
18		eeanv 2182	. . . 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 865	. . . . . 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 17301	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
25	24	3expb 1266	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
26	21, 25	sylan 488	. . . . . . . . 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 3603	. . . . . . . . . . 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 753	. . . . . . . . . 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 796	. . . . . . . . . 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 17762	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
34	30, 31, 33	syl2anc 693	. . . . . . . . 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 17302	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
36	21, 35	sylan 488	. . . . . . . . 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 475	. . . . . . . . . . 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 11723	. . . . . . . . . 10 ⊢ ℕ = (ℤ_≥‘1)
40	38, 39	syl6eleq 2711	. . . . . . . . 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		frn 6053	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
44	42, 43	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) → ran 𝐹 ⊆ 𝐵)
45		simprr 796	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) → 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)
46		f1ocnv 6149	. . . . . . . . . . 11 ⊢ (𝑔:(1...(#‘𝑊))–1-1-onto→𝑊 → ^◡𝑔:𝑊–1-1-onto→(1...(#‘𝑊)))
47	45, 46	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)))
48		simprl 794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) → 𝑓:(1...(#‘𝑊))–1-1-onto→𝑊)
49		f1oco 6159	. . . . . . . . . 10 ⊢ ((^◡𝑔:𝑊–1-1-onto→(1...(#‘𝑊)) ∧ 𝑓:(1...(#‘𝑊))–1-1-onto→𝑊) → (^◡𝑔 ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(1...(#‘𝑊)))
50	47, 48, 49	syl2anc 693	. . . . . . . . 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)))
51		f1of 6137	. . . . . . . . . . . 12 ⊢ (𝑔:(1...(#‘𝑊))–1-1-onto→𝑊 → 𝑔:(1...(#‘𝑊))⟶𝑊)
52	45, 51	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) → 𝑔:(1...(#‘𝑊))⟶𝑊)
53		fvco3 6275	. . . . . . . . . . 11 ⊢ ((𝑔:(1...(#‘𝑊))⟶𝑊 ∧ 𝑥 ∈ (1...(#‘𝑊))) → ((𝐹 ∘ 𝑔)‘𝑥) = (𝐹‘(𝑔‘𝑥)))
54	52, 53	sylan 488	. . . . . . . . . 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)))
55		ffn 6045	. . . . . . . . . . . . 13 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
56	42, 55	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) → 𝐹 Fn 𝐴)
57	56	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (1...(#‘𝑊))) → 𝐹 Fn 𝐴)
58		gsumval3a.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊 ⊆ 𝐴)
59	58	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) → 𝑊 ⊆ 𝐴)
60	52, 59	fssd 6057	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) → 𝑔:(1...(#‘𝑊))⟶𝐴)
61	60	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (1...(#‘𝑊))) → (𝑔‘𝑥) ∈ 𝐴)
62		fnfvelrn 6356	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐴) → (𝐹‘(𝑔‘𝑥)) ∈ ran 𝐹)
63	57, 61, 62	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (1...(#‘𝑊))) → (𝐹‘(𝑔‘𝑥)) ∈ ran 𝐹)
64	54, 63	eqeltrd 2701	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (1...(#‘𝑊))) → ((𝐹 ∘ 𝑔)‘𝑥) ∈ ran 𝐹)
65		f1of 6137	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 → 𝑓:(1...(#‘𝑊))⟶𝑊)
66	48, 65	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) → 𝑓:(1...(#‘𝑊))⟶𝑊)
67		fvco3 6275	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(#‘𝑊))⟶𝑊 ∧ 𝑘 ∈ (1...(#‘𝑊))) → ((^◡𝑔 ∘ 𝑓)‘𝑘) = (^◡𝑔‘(𝑓‘𝑘)))
68	66, 67	sylan 488	. . . . . . . . . . . . 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)))
69	68	fveq2d 6195	. . . . . . . . . . . 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))))
70	45	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)
71	66	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → (𝑓‘𝑘) ∈ 𝑊)
72		f1ocnvfv2 6533	. . . . . . . . . . . . 13 ⊢ ((𝑔:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ (𝑓‘𝑘) ∈ 𝑊) → (𝑔‘(^◡𝑔‘(𝑓‘𝑘))) = (𝑓‘𝑘))
73	70, 71, 72	syl2anc 693	. . . . . . . . . . . 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))
74	69, 73	eqtr2d 2657	. . . . . . . . . . 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)))
75	74	fveq2d 6195	. . . . . . . . . 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))))
76		fvco3 6275	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(#‘𝑊))⟶𝑊 ∧ 𝑘 ∈ (1...(#‘𝑊))) → ((𝐹 ∘ 𝑓)‘𝑘) = (𝐹‘(𝑓‘𝑘)))
77	66, 76	sylan 488	. . . . . . . . . 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)))
78		f1of 6137	. . . . . . . . . . . . 13 ⊢ ((^◡𝑔 ∘ 𝑓):(1...(#‘𝑊))–1-1-onto→(1...(#‘𝑊)) → (^◡𝑔 ∘ 𝑓):(1...(#‘𝑊))⟶(1...(#‘𝑊)))
79	50, 78	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) → (^◡𝑔 ∘ 𝑓):(1...(#‘𝑊))⟶(1...(#‘𝑊)))
80	79	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → ((^◡𝑔 ∘ 𝑓)‘𝑘) ∈ (1...(#‘𝑊)))
81		fvco3 6275	. . . . . . . . . . . 12 ⊢ ((𝑔:(1...(#‘𝑊))⟶𝐴 ∧ ((^◡𝑔 ∘ 𝑓)‘𝑘) ∈ (1...(#‘𝑊))) → ((𝐹 ∘ 𝑔)‘((^◡𝑔 ∘ 𝑓)‘𝑘)) = (𝐹‘(𝑔‘((^◡𝑔 ∘ 𝑓)‘𝑘))))
82	60, 81	sylan 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) ∧ ((^◡𝑔 ∘ 𝑓)‘𝑘) ∈ (1...(#‘𝑊))) → ((𝐹 ∘ 𝑔)‘((^◡𝑔 ∘ 𝑓)‘𝑘)) = (𝐹‘(𝑔‘((^◡𝑔 ∘ 𝑓)‘𝑘))))
83	80, 82	syldan 487	. . . . . . . . . 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))))
84	75, 77, 83	3eqtr4d 2666	. . . . . . . . 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)))
85	26, 34, 36, 40, 44, 50, 64, 84	seqf1o 12842	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊)) → (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘𝑊)))
86		eqeq12 2635	. . . . . . . 8 ⊢ ((𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘𝑊))) → (𝑥 = 𝑦 ↔ (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘𝑊))))
87	85, 86	syl5ibrcom 237	. . . . . . 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))
88	87	expimpd 629	. . . . . 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))
89	19, 88	syl5bir 233	. . . . 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))
90	89	exlimdvv 1862	. . . 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))
91	18, 90	syl5bir 233	. . 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))
92	91	alrimivv 1856	. 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))
93		eqeq1 2626	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊))))
94	93	anbi2d 740	. . . . 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)))))
95	94	exbidv 1850	. . . 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)))))
96		f1oeq1 6127	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓:(1...(#‘𝑊))–1-1-onto→𝑊 ↔ 𝑔:(1...(#‘𝑊))–1-1-onto→𝑊))
97		coeq2 5280	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔))
98	97	seqeq3d 12809	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → seq1( + , (𝐹 ∘ 𝑓)) = seq1( + , (𝐹 ∘ 𝑔)))
99	98	fveq1d 6193	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘𝑊)))
100	99	eqeq2d 2632	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(#‘𝑊))))
101	96, 100	anbi12d 747	. . . . 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)))))
102	101	cbvexv 2275	. . . 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))))
103	95, 102	syl6bb 276	. . 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)))))
104	103	eu4 2518	. 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)))
105	17, 92, 104	sylanbrc 698	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 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃!weu 2470 ≠ wne 2794 ⊆ wss 3574 ∅c0 3915 ^◡ccnv 5113 ran crn 5115 ∘ ccom 5118 Fn wfn 5883 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 1c1 9937 ℕcn 11020 ℤ_≥cuz 11687 ...cfz 12326 seqcseq 12801 #chash 13117 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 Mndcmnd 17294 Cntzccntz 17748

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-cntz 17750

This theorem is referenced by: gsumval3lem2 18307

W3C validator