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

Theorem gsumval3eu 19681
Description: The group sum as defined in gsumval3a 19680 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 2948 . . . . 5 (𝜑 → ¬ 𝑊 = ∅)
3 gsumval3a.t . . . . . . 7 (𝜑𝑊 ∈ Fin)
4 fz1f1o 15595 . . . . . . 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 862 . . . . 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 496 . . 3 (𝜑 → ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
9 excom 2162 . . . 4 (∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
10 exancom 1864 . . . . . 6 (∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑥(𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊))
11 fvex 6855 . . . . . . 7 (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∈ V
12 biidd 261 . . . . . . 7 (𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) → (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑓:(1...(♯‘𝑊))–1-1-onto𝑊))
1311, 12ceqsexv 3494 . . . . . 6 (∃𝑥(𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊) ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
1410, 13bitri 274 . . . . 5 (∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
1514exbii 1850 . . . 4 (∃𝑓𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
169, 15bitri 274 . . 3 (∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
178, 16sylibr 233 . 2 (𝜑 → ∃𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
18 exdistrv 1959 . . . 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 654 . . . . . 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 18564 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
25243expb 1120 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
2621, 25sylan 580 . . . . . . . . 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 3944 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ ran 𝐹) → 𝑥 ∈ (𝑍‘ran 𝐹))
3029adantrr 715 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → 𝑥 ∈ (𝑍‘ran 𝐹))
31 simprr 771 . . . . . . . . . 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 19109 . . . . . . . . . 10 ((𝑥 ∈ (𝑍‘ran 𝐹) ∧ 𝑦 ∈ ran 𝐹) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3430, 31, 33syl2anc 584 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐹)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3522, 23mndass 18565 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
3621, 35sylan 580 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
377simpld 495 . . . . . . . . . . 11 (𝜑 → (♯‘𝑊) ∈ ℕ)
3837adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (♯‘𝑊) ∈ ℕ)
39 nnuz 12806 . . . . . . . . . 10 ℕ = (ℤ‘1)
4038, 39eleqtrdi 2848 . . . . . . . . 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)
4342frnd 6676 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → ran 𝐹𝐵)
44 simprr 771 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)
45 f1ocnv 6796 . . . . . . . . . . 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 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊)
48 f1oco 6807 . . . . . . . . . 10 ((𝑔:𝑊1-1-onto→(1...(♯‘𝑊)) ∧ 𝑓:(1...(♯‘𝑊))–1-1-onto𝑊) → (𝑔𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)))
4946, 47, 48syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝑔𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)))
50 f1of 6784 . . . . . . . . . . . 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 6940 . . . . . . . . . . 11 ((𝑔:(1...(♯‘𝑊))⟶𝑊𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) = (𝐹‘(𝑔𝑥)))
5351, 52sylan 580 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) = (𝐹‘(𝑔𝑥)))
5442ffnd 6669 . . . . . . . . . . 11 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝐹 Fn 𝐴)
55 gsumval3a.s . . . . . . . . . . . . . 14 (𝜑𝑊𝐴)
5655adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑊𝐴)
5751, 56fssd 6686 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝐴)
5857ffvelcdmda 7035 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝑔𝑥) ∈ 𝐴)
59 fnfvelrn 7031 . . . . . . . . . . 11 ((𝐹 Fn 𝐴 ∧ (𝑔𝑥) ∈ 𝐴) → (𝐹‘(𝑔𝑥)) ∈ ran 𝐹)
6054, 58, 59syl2an2r 683 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑔𝑥)) ∈ ran 𝐹)
6153, 60eqeltrd 2838 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘𝑥) ∈ ran 𝐹)
62 f1of 6784 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑓:(1...(♯‘𝑊))⟶𝑊)
6347, 62syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → 𝑓:(1...(♯‘𝑊))⟶𝑊)
64 fvco3 6940 . . . . . . . . . . . . . 14 ((𝑓:(1...(♯‘𝑊))⟶𝑊𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) = (𝑔‘(𝑓𝑘)))
6563, 64sylan 580 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) = (𝑔‘(𝑓𝑘)))
6665fveq2d 6846 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘((𝑔𝑓)‘𝑘)) = (𝑔‘(𝑔‘(𝑓𝑘))))
6763ffvelcdmda 7035 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓𝑘) ∈ 𝑊)
68 f1ocnvfv2 7223 . . . . . . . . . . . . 13 ((𝑔:(1...(♯‘𝑊))–1-1-onto𝑊 ∧ (𝑓𝑘) ∈ 𝑊) → (𝑔‘(𝑔‘(𝑓𝑘))) = (𝑓𝑘))
6944, 67, 68syl2an2r 683 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑔‘(𝑔‘(𝑓𝑘))) = (𝑓𝑘))
7066, 69eqtr2d 2777 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓𝑘) = (𝑔‘((𝑔𝑓)‘𝑘)))
7170fveq2d 6846 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝐹‘(𝑓𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
72 fvco3 6940 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝑊))⟶𝑊𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = (𝐹‘(𝑓𝑘)))
7363, 72sylan 580 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = (𝐹‘(𝑓𝑘)))
74 f1of 6784 . . . . . . . . . . . . 13 ((𝑔𝑓):(1...(♯‘𝑊))–1-1-onto→(1...(♯‘𝑊)) → (𝑔𝑓):(1...(♯‘𝑊))⟶(1...(♯‘𝑊)))
7549, 74syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (𝑔𝑓):(1...(♯‘𝑊))⟶(1...(♯‘𝑊)))
7675ffvelcdmda 7035 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝑔𝑓)‘𝑘) ∈ (1...(♯‘𝑊)))
77 fvco3 6940 . . . . . . . . . . 11 ((𝑔:(1...(♯‘𝑊))⟶𝐴 ∧ ((𝑔𝑓)‘𝑘) ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
7857, 76, 77syl2an2r 683 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)) = (𝐹‘(𝑔‘((𝑔𝑓)‘𝑘))))
7971, 73, 783eqtr4d 2786 . . . . . . . . 9 (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → ((𝐹𝑓)‘𝑘) = ((𝐹𝑔)‘((𝑔𝑓)‘𝑘)))
8026, 34, 36, 40, 43, 49, 61, 79seqf1o 13949 . . . . . . . 8 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))
81 eqeq12 2753 . . . . . . . 8 ((𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))) → (𝑥 = 𝑦 ↔ (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))))
8280, 81syl5ibrcom 246 . . . . . . 7 ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊)) → ((𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))) → 𝑥 = 𝑦))
8382expimpd 454 . . . . . 6 (𝜑 → (((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊) ∧ (𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ∧ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8419, 83biimtrrid 242 . . . . 5 (𝜑 → (((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8584exlimdvv 1937 . . . 4 (𝜑 → (∃𝑓𝑔((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8618, 85biimtrrid 242 . . 3 (𝜑 → ((∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ∧ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))) → 𝑥 = 𝑦))
8786alrimivv 1931 . 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))
88 eqeq1 2740 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))
8988anbi2d 629 . . . . 5 (𝑥 = 𝑦 → ((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))
9089exbidv 1924 . . . 4 (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))
91 f1oeq1 6772 . . . . . 6 (𝑓 = 𝑔 → (𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑔:(1...(♯‘𝑊))–1-1-onto𝑊))
92 coeq2 5814 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐹𝑓) = (𝐹𝑔))
9392seqeq3d 13914 . . . . . . . 8 (𝑓 = 𝑔 → seq1( + , (𝐹𝑓)) = seq1( + , (𝐹𝑔)))
9493fveq1d 6844 . . . . . . 7 (𝑓 = 𝑔 → (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))
9594eqeq2d 2747 . . . . . 6 (𝑓 = 𝑔 → (𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))))
9691, 95anbi12d 631 . . . . 5 (𝑓 = 𝑔 → ((𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ (𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))))
9796cbvexvw 2040 . . . 4 (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊))))
9890, 97bitrdi 286 . . 3 (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))) ↔ ∃𝑔(𝑔:(1...(♯‘𝑊))–1-1-onto𝑊𝑦 = (seq1( + , (𝐹𝑔))‘(♯‘𝑊)))))
9998eu4 2615 . 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)))
10017, 87, 99sylanbrc 583 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 396  wo 845  w3a 1087  wal 1539   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2566  wne 2943  wss 3910  c0 4282  ccnv 5632  ran crn 5634  ccom 5637   Fn wfn 6491  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  Fincfn 8883  1c1 11052  cn 12153  cuz 12763  ...cfz 13424  seqcseq 13906  chash 14230  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Mndcmnd 18556  Cntzccntz 19095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-cntz 19097
This theorem is referenced by:  gsumval3lem2  19683
  Copyright terms: Public domain W3C validator