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Theorem gsumval3eu 18221
Description: The group sum as defined in gsumval3a 18220 is uniquely defined. (Contributed by Mario Carneiro, 8-Dec-2014.)
Hypotheses
Ref Expression
gsumval3.b 𝐵 = (Base‘𝐺)
gsumval3.0 0 = (0g𝐺)
gsumval3.p + = (+g𝐺)
gsumval3.z 𝑍 = (Cntz‘𝐺)
gsumval3.g (𝜑𝐺 ∈ Mnd)
gsumval3.a (𝜑𝐴𝑉)
gsumval3.f (𝜑𝐹:𝐴𝐵)
gsumval3.c (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumval3a.t (𝜑𝑊 ∈ Fin)
gsumval3a.n (𝜑𝑊 ≠ ∅)
gsumval3a.s (𝜑𝑊𝐴)
Assertion
Ref Expression
gsumval3eu (𝜑 → ∃!𝑥𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘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.
StepHypRef Expression
1 gsumval3a.n . . . . . 6 (𝜑𝑊 ≠ ∅)
21neneqd 2801 . . . . 5 (𝜑 → ¬ 𝑊 = ∅)
3 gsumval3a.t . . . . . . 7 (𝜑𝑊 ∈ Fin)
4 fz1f1o 14369 . . . . . . 7 (𝑊 ∈ Fin → (𝑊 = ∅ ∨ ((#‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝑊))–1-1-onto𝑊)))
53, 4syl 17 . . . . . 6 (𝜑 → (𝑊 = ∅ ∨ ((#‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝑊))–1-1-onto𝑊)))
65ord 392 . . . . 5 (𝜑 → (¬ 𝑊 = ∅ → ((#‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝑊))–1-1-onto𝑊)))
72, 6mpd 15 . . . 4 (𝜑 → ((#‘𝑊) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝑊))–1-1-onto𝑊))
87simprd 479 . . 3 (𝜑 → ∃𝑓 𝑓:(1...(#‘𝑊))–1-1-onto𝑊)
9 excom 2044 . . . 4 (∃𝑥𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ↔ ∃𝑓𝑥(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))))
10 exancom 1785 . . . . . 6 (∃𝑥(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ↔ ∃𝑥(𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)) ∧ 𝑓:(1...(#‘𝑊))–1-1-onto𝑊))
11 fvex 6160 . . . . . . 7 (seq1( + , (𝐹𝑓))‘(#‘𝑊)) ∈ V
12 biidd 252 . . . . . . 7 (𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)) → (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑓:(1...(#‘𝑊))–1-1-onto𝑊))
1311, 12ceqsexv 3233 . . . . . 6 (∃𝑥(𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)) ∧ 𝑓:(1...(#‘𝑊))–1-1-onto𝑊) ↔ 𝑓:(1...(#‘𝑊))–1-1-onto𝑊)
1410, 13bitri 264 . . . . 5 (∃𝑥(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ↔ 𝑓:(1...(#‘𝑊))–1-1-onto𝑊)
1514exbii 1772 . . . 4 (∃𝑓𝑥(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ↔ ∃𝑓 𝑓:(1...(#‘𝑊))–1-1-onto𝑊)
169, 15bitri 264 . . 3 (∃𝑥𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ↔ ∃𝑓 𝑓:(1...(#‘𝑊))–1-1-onto𝑊)
178, 16sylibr 224 . 2 (𝜑 → ∃𝑥𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))))
18 eeanv 2186 . . . 4 (∃𝑓𝑔((𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ∧ (𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))) ↔ (∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ∧ ∃𝑔(𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))))
19 an4 864 . . . . . 6 (((𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊) ∧ (𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))) ↔ ((𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ∧ (𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))))
20 gsumval3.g . . . . . . . . . . 11 (𝜑𝐺 ∈ Mnd)
2120adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → 𝐺 ∈ Mnd)
22 gsumval3.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐺)
23 gsumval3.p . . . . . . . . . . . 12 + = (+g𝐺)
2422, 23mndcl 17217 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
25243expb 1263 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
2621, 25sylan 488 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
27 gsumval3.c . . . . . . . . . . . . 13 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
2827adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
2928sselda 3588 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ ran 𝐹) → 𝑥 ∈ (𝑍‘ran 𝐹))
3029adantrr 752 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → 𝑥 ∈ (𝑍‘ran 𝐹))
31 simprr 795 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → 𝑦 ∈ ran 𝐹)
32 gsumval3.z . . . . . . . . . . 11 𝑍 = (Cntz‘𝐺)
3323, 32cntzi 17678 . . . . . . . . . 10 ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3430, 31, 33syl2anc 692 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3522, 23mndass 17218 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
3621, 35sylan 488 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
377simpld 475 . . . . . . . . . . 11 (𝜑 → (#‘𝑊) ∈ ℕ)
3837adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → (#‘𝑊) ∈ ℕ)
39 nnuz 11667 . . . . . . . . . 10 ℕ = (ℤ‘1)
4038, 39syl6eleq 2714 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → (#‘𝑊) ∈ (ℤ‘1))
41 gsumval3.f . . . . . . . . . . 11 (𝜑𝐹:𝐴𝐵)
4241adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → 𝐹:𝐴𝐵)
43 frn 6012 . . . . . . . . . 10 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
4442, 43syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → ran 𝐹𝐵)
45 simprr 795 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(#‘𝑊))–1-1-onto𝑊)
46 f1ocnv 6108 . . . . . . . . . . 11 (𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑔:𝑊1-1-onto→(1...(#‘𝑊)))
4745, 46syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → 𝑔:𝑊1-1-onto→(1...(#‘𝑊)))
48 simprl 793 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(#‘𝑊))–1-1-onto𝑊)
49 f1oco 6118 . . . . . . . . . 10 ((𝑔:𝑊1-1-onto→(1...(#‘𝑊)) ∧ 𝑓:(1...(#‘𝑊))–1-1-onto𝑊) → (𝑔𝑓):(1...(#‘𝑊))–1-1-onto→(1...(#‘𝑊)))
5047, 48, 49syl2anc 692 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → (𝑔𝑓):(1...(#‘𝑊))–1-1-onto→(1...(#‘𝑊)))
51 f1of 6096 . . . . . . . . . . . 12 (𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))⟶𝑊)
5245, 51syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(#‘𝑊))⟶𝑊)
53 fvco3 6233 . . . . . . . . . . 11 ((𝑔:(1...(#‘𝑊))⟶𝑊𝑥 ∈ (1...(#‘𝑊))) → ((𝐹𝑔)‘𝑥) = (𝐹‘(𝑔𝑥)))
5452, 53sylan 488 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(#‘𝑊))) → ((𝐹𝑔)‘𝑥) = (𝐹‘(𝑔𝑥)))
55 ffn 6004 . . . . . . . . . . . . 13 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
5642, 55syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → 𝐹 Fn 𝐴)
5756adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(#‘𝑊))) → 𝐹 Fn 𝐴)
58 gsumval3a.s . . . . . . . . . . . . . 14 (𝜑𝑊𝐴)
5958adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → 𝑊𝐴)
6052, 59fssd 6016 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(#‘𝑊))⟶𝐴)
6160ffvelrnda 6316 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(#‘𝑊))) → (𝑔𝑥) ∈ 𝐴)
62 fnfvelrn 6313 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ (𝑔𝑥) ∈ 𝐴) → (𝐹‘(𝑔𝑥)) ∈ ran 𝐹)
6357, 61, 62syl2anc 692 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(#‘𝑊))) → (𝐹‘(𝑔𝑥)) ∈ ran 𝐹)
6454, 63eqeltrd 2704 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(#‘𝑊))) → ((𝐹𝑔)‘𝑥) ∈ ran 𝐹)
65 f1of 6096 . . . . . . . . . . . . . . 15 (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑓:(1...(#‘𝑊))⟶𝑊)
6648, 65syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(#‘𝑊))⟶𝑊)
67 fvco3 6233 . . . . . . . . . . . . . 14 ((𝑓:(1...(#‘𝑊))⟶𝑊𝑘 ∈ (1...(#‘𝑊))) → ((𝑔𝑓)‘𝑘) = (𝑔‘(𝑓𝑘)))
6866, 67sylan 488 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → ((𝑔𝑓)‘𝑘) = (𝑔‘(𝑓𝑘)))
6968fveq2d 6154 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → (𝑔‘((𝑔𝑓)‘𝑘)) = (𝑔‘(𝑔‘(𝑓𝑘))))
7045adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → 𝑔:(1...(#‘𝑊))–1-1-onto𝑊)
7166ffvelrnda 6316 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → (𝑓𝑘) ∈ 𝑊)
72 f1ocnvfv2 6488 . . . . . . . . . . . . 13 ((𝑔:(1...(#‘𝑊))–1-1-onto𝑊 ∧ (𝑓𝑘) ∈ 𝑊) → (𝑔‘(𝑔‘(𝑓𝑘))) = (𝑓𝑘))
7370, 71, 72syl2anc 692 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → (𝑔‘(𝑔‘(𝑓𝑘))) = (𝑓𝑘))
7469, 73eqtr2d 2661 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → (𝑓𝑘) = (𝑔‘((𝑔𝑓)‘𝑘)))
7574fveq2d 6154 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → (𝐹‘(𝑓𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
76 fvco3 6233 . . . . . . . . . . 11 ((𝑓:(1...(#‘𝑊))⟶𝑊𝑘 ∈ (1...(#‘𝑊))) → ((𝐹𝑓)‘𝑘) = (𝐹‘(𝑓𝑘)))
7766, 76sylan 488 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → ((𝐹𝑓)‘𝑘) = (𝐹‘(𝑓𝑘)))
78 f1of 6096 . . . . . . . . . . . . 13 ((𝑔𝑓):(1...(#‘𝑊))–1-1-onto→(1...(#‘𝑊)) → (𝑔𝑓):(1...(#‘𝑊))⟶(1...(#‘𝑊)))
7950, 78syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → (𝑔𝑓):(1...(#‘𝑊))⟶(1...(#‘𝑊)))
8079ffvelrnda 6316 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → ((𝑔𝑓)‘𝑘) ∈ (1...(#‘𝑊)))
81 fvco3 6233 . . . . . . . . . . . 12 ((𝑔:(1...(#‘𝑊))⟶𝐴 ∧ ((𝑔𝑓)‘𝑘) ∈ (1...(#‘𝑊))) → ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
8260, 81sylan 488 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ ((𝑔𝑓)‘𝑘) ∈ (1...(#‘𝑊))) → ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
8380, 82syldan 487 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
8475, 77, 833eqtr4d 2670 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(#‘𝑊))) → ((𝐹𝑓)‘𝑘) = ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)))
8526, 34, 36, 40, 44, 50, 64, 84seqf1o 12779 . . . . . . . 8 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → (seq1( + , (𝐹𝑓))‘(#‘𝑊)) = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))
86 eqeq12 2639 . . . . . . . 8 ((𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊))) → (𝑥 = 𝑦 ↔ (seq1( + , (𝐹𝑓))‘(#‘𝑊)) = (seq1( + , (𝐹𝑔))‘(#‘𝑊))))
8785, 86syl5ibrcom 237 . . . . . . 7 ((𝜑 ∧ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊)) → ((𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊))) → 𝑥 = 𝑦))
8887expimpd 628 . . . . . 6 (𝜑 → (((𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊) ∧ (𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))) → 𝑥 = 𝑦))
8919, 88syl5bir 233 . . . . 5 (𝜑 → (((𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ∧ (𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))) → 𝑥 = 𝑦))
9089exlimdvv 1864 . . . 4 (𝜑 → (∃𝑓𝑔((𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ∧ (𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))) → 𝑥 = 𝑦))
9118, 90syl5bir 233 . . 3 (𝜑 → ((∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ∧ ∃𝑔(𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))) → 𝑥 = 𝑦))
9291alrimivv 1858 . 2 (𝜑 → ∀𝑥𝑦((∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ∧ ∃𝑔(𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))) → 𝑥 = 𝑦))
93 eqeq1 2630 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))))
9493anbi2d 739 . . . . 5 (𝑥 = 𝑦 → ((𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ↔ (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)))))
9594exbidv 1852 . . . 4 (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ↔ ∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)))))
96 f1oeq1 6086 . . . . . 6 (𝑓 = 𝑔 → (𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑔:(1...(#‘𝑊))–1-1-onto𝑊))
97 coeq2 5245 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐹𝑓) = (𝐹𝑔))
9897seqeq3d 12746 . . . . . . . 8 (𝑓 = 𝑔 → seq1( + , (𝐹𝑓)) = seq1( + , (𝐹𝑔)))
9998fveq1d 6152 . . . . . . 7 (𝑓 = 𝑔 → (seq1( + , (𝐹𝑓))‘(#‘𝑊)) = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))
10099eqeq2d 2636 . . . . . 6 (𝑓 = 𝑔 → (𝑦 = (seq1( + , (𝐹𝑓))‘(#‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊))))
10196, 100anbi12d 746 . . . . 5 (𝑓 = 𝑔 → ((𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ↔ (𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))))
102101cbvexv 2279 . . . 4 (∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ↔ ∃𝑔(𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊))))
10395, 102syl6bb 276 . . 3 (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ↔ ∃𝑔(𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))))
104103eu4 2522 . 2 (∃!𝑥𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ↔ (∃𝑥𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ∧ ∀𝑥𝑦((∃𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))) ∧ ∃𝑔(𝑔:(1...(#‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(#‘𝑊)))) → 𝑥 = 𝑦)))
10517, 92, 104sylanbrc 697 1 (𝜑 → ∃!𝑥𝑓(𝑓:(1...(#‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(#‘𝑊))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1992  ∃!weu 2474  wne 2796  wss 3560  c0 3896  ccnv 5078  ran crn 5080  ccom 5083   Fn wfn 5845  wf 5846  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  Fincfn 7900  1c1 9882  cn 10965  cuz 11631  ...cfz 12265  seqcseq 12738  #chash 13054  Basecbs 15776  +gcplusg 15857  0gc0g 16016  Mndcmnd 17210  Cntzccntz 17664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-seq 12739  df-hash 13055  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-cntz 17666
This theorem is referenced by:  gsumval3lem2  18223
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