MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumval3eu Structured version   Visualization version   GIF version

Theorem gsumval3eu 18502
Description: The group sum as defined in gsumval3a 18501 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 2983 . . . . 5 (𝜑 → ¬ 𝑊 = ∅)
3 gsumval3a.t . . . . . . 7 (𝜑𝑊 ∈ Fin)
4 fz1f1o 14660 . . . . . . 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 882 . . . . 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 485 . . 3 (𝜑 → ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
9 excom 2209 . . . 4 (∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
10 exancom 1947 . . . . . 6 (∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑥(𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊))
11 fvex 6417 . . . . . . 7 (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∈ V
12 biidd 253 . . . . . . 7 (𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) → (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑓:(1...(♯‘𝑊))–1-1-onto𝑊))
1311, 12ceqsexv 3436 . . . . . 6 (∃𝑥(𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊) ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
1410, 13bitri 266 . . . . 5 (∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
1514exbii 1933 . . . 4 (∃𝑓𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
169, 15bitri 266 . . 3 (∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
178, 16sylibr 225 . 2 (𝜑 → ∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
18 eeanv 2356 . . . 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 638 . . . . . 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 468 . . . . . . . . . 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 17502 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
25243expb 1142 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
2621, 25sylan 571 . . . . . . . . 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 468 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
2928sselda 3798 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ ran 𝐹) → 𝑥 ∈ (𝑍‘ran 𝐹))
3029adantrr 699 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → 𝑥 ∈ (𝑍‘ran 𝐹))
31 simprr 780 . . . . . . . . . 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 17959 . . . . . . . . . 10 ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3430, 31, 33syl2anc 575 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3522, 23mndass 17503 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
3621, 35sylan 571 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
377simpld 484 . . . . . . . . . . 11 (𝜑 → (♯‘𝑊) ∈ ℕ)
3837adantr 468 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (♯‘𝑊) ∈ ℕ)
39 nnuz 11937 . . . . . . . . . 10 ℕ = (ℤ‘1)
4038, 39syl6eleq 2895 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (♯‘𝑊) ∈ (ℤ‘1))
41 gsumval3.f . . . . . . . . . . 11 (𝜑𝐹:𝐴𝐵)
4241adantr 468 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐹:𝐴𝐵)
4342frnd 6259 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran 𝐹𝐵)
44 simprr 780 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)
45 f1ocnv 6361 . . . . . . . . . . 11 (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:𝑊1-1-onto→(1...(♯‘𝑊)))
4644, 45syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:𝑊1-1-onto→(1...(♯‘𝑊)))
47 simprl 778 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
48 f1oco 6371 . . . . . . . . . 10 ((𝑔:𝑊1-1-onto→(1...(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊) → (𝑔𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)))
4946, 47, 48syl2anc 575 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝑔𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)))
50 f1of 6349 . . . . . . . . . . . 12 (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))⟶𝑊)
5144, 50syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝑊)
52 fvco3 6492 . . . . . . . . . . 11 ((𝑔:(1...(♯‘𝑊))⟶𝑊𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) = (𝐹‘(𝑔𝑥)))
5351, 52sylan 571 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) = (𝐹‘(𝑔𝑥)))
5442ffnd 6253 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐹 Fn 𝐴)
5554adantr 468 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → 𝐹 Fn 𝐴)
56 gsumval3a.s . . . . . . . . . . . . . 14 (𝜑𝑊𝐴)
5756adantr 468 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑊𝐴)
5851, 57fssd 6266 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝐴)
5958ffvelrnda 6577 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝑔𝑥) ∈ 𝐴)
60 fnfvelrn 6574 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ (𝑔𝑥) ∈ 𝐴) → (𝐹‘(𝑔𝑥)) ∈ ran 𝐹)
6155, 59, 60syl2anc 575 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑔𝑥)) ∈ ran 𝐹)
6253, 61eqeltrd 2885 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) ∈ ran 𝐹)
63 f1of 6349 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑓:(1...(♯‘𝑊))⟶𝑊)
6447, 63syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))⟶𝑊)
65 fvco3 6492 . . . . . . . . . . . . . 14 ((𝑓:(1...(♯‘𝑊))⟶𝑊𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) = (𝑔‘(𝑓𝑘)))
6664, 65sylan 571 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) = (𝑔‘(𝑓𝑘)))
6766fveq2d 6408 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘((𝑔𝑓)‘𝑘)) = (𝑔‘(𝑔‘(𝑓𝑘))))
6844adantr 468 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → 𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)
6964ffvelrnda 6577 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓𝑘) ∈ 𝑊)
70 f1ocnvfv2 6753 . . . . . . . . . . . . 13 ((𝑔:(1...(♯‘𝑊))–1-1-onto𝑊 ∧ (𝑓𝑘) ∈ 𝑊) → (𝑔‘(𝑔‘(𝑓𝑘))) = (𝑓𝑘))
7168, 69, 70syl2anc 575 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘(𝑔‘(𝑓𝑘))) = (𝑓𝑘))
7267, 71eqtr2d 2841 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓𝑘) = (𝑔‘((𝑔𝑓)‘𝑘)))
7372fveq2d 6408 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑓𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
74 fvco3 6492 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝑊))⟶𝑊𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = (𝐹‘(𝑓𝑘)))
7564, 74sylan 571 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = (𝐹‘(𝑓𝑘)))
76 f1of 6349 . . . . . . . . . . . . 13 ((𝑔𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)) → (𝑔𝑓):(1...(♯‘𝑊))⟶(1...(♯‘𝑊)))
7749, 76syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝑔𝑓):(1...(♯‘𝑊))⟶(1...(♯‘𝑊)))
7877ffvelrnda 6577 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) ∈ (1...(♯‘𝑊)))
79 fvco3 6492 . . . . . . . . . . . 12 ((𝑔:(1...(♯‘𝑊))⟶𝐴 ∧ ((𝑔𝑓)‘𝑘) ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
8058, 79sylan 571 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ ((𝑔𝑓)‘𝑘) ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
8178, 80syldan 581 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
8273, 75, 813eqtr4d 2850 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)))
8326, 34, 36, 40, 43, 49, 62, 82seqf1o 13061 . . . . . . . 8 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))
84 eqeq12 2819 . . . . . . . 8 ((𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))) → (𝑥 = 𝑦 ↔ (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))))
8583, 84syl5ibrcom 238 . . . . . . 7 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))) → 𝑥 = 𝑦))
8685expimpd 443 . . . . . 6 (𝜑 → (((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊) ∧ (𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8719, 86syl5bir 234 . . . . 5 (𝜑 → (((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8887exlimdvv 2025 . . . 4 (𝜑 → (∃𝑓𝑔((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8918, 88syl5bir 234 . . 3 (𝜑 → ((∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
9089alrimivv 2019 . 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))
91 eqeq1 2810 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
9291anbi2d 616 . . . . 5 (𝑥 = 𝑦 → ((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))
9392exbidv 2012 . . . 4 (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))
94 f1oeq1 6339 . . . . . 6 (𝑓 = 𝑔 → (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊))
95 coeq2 5482 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐹𝑓) = (𝐹𝑔))
9695seqeq3d 13028 . . . . . . . 8 (𝑓 = 𝑔 → seq1( + , (𝐹𝑓)) = seq1( + , (𝐹𝑔)))
9796fveq1d 6406 . . . . . . 7 (𝑓 = 𝑔 → (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))
9897eqeq2d 2816 . . . . . 6 (𝑓 = 𝑔 → (𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))))
9994, 98anbi12d 618 . . . . 5 (𝑓 = 𝑔 → ((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))))
10099cbvexvw 2137 . . . 4 (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))))
10193, 100syl6bb 278 . . 3 (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))))
102101eu4 2681 . 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)))
10317, 90, 102sylanbrc 574 1 (𝜑 → ∃!𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 865  w3a 1100  wal 1635   = wceq 1637  wex 1859  wcel 2156  ∃!weu 2630  wne 2978  wss 3769  c0 4116  ccnv 5310  ran crn 5312  ccom 5315   Fn wfn 6092  wf 6093  1-1-ontowf1o 6096  cfv 6097  (class class class)co 6870  Fincfn 8188  1c1 10218  cn 11301  cuz 11900  ...cfz 12545  seqcseq 13020  chash 13333  Basecbs 16064  +gcplusg 16149  0gc0g 16301  Mndcmnd 17495  Cntzccntz 17945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-cnex 10273  ax-resscn 10274  ax-1cn 10275  ax-icn 10276  ax-addcl 10277  ax-addrcl 10278  ax-mulcl 10279  ax-mulrcl 10280  ax-mulcom 10281  ax-addass 10282  ax-mulass 10283  ax-distr 10284  ax-i2m1 10285  ax-1ne0 10286  ax-1rid 10287  ax-rnegex 10288  ax-rrecex 10289  ax-cnre 10290  ax-pre-lttri 10291  ax-pre-lttrn 10292  ax-pre-ltadd 10293  ax-pre-mulgt0 10294
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1st 7394  df-2nd 7395  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-oadd 7796  df-er 7975  df-en 8189  df-dom 8190  df-sdom 8191  df-fin 8192  df-card 9044  df-pnf 10357  df-mnf 10358  df-xr 10359  df-ltxr 10360  df-le 10361  df-sub 10549  df-neg 10550  df-nn 11302  df-n0 11556  df-z 11640  df-uz 11901  df-fz 12546  df-fzo 12686  df-seq 13021  df-hash 13334  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-cntz 17947
This theorem is referenced by:  gsumval3lem2  18504
  Copyright terms: Public domain W3C validator