| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | gsumval3a.n | . . . . . 6
⊢ (𝜑 → 𝑊 ≠ ∅) | 
| 2 | 1 | neneqd 2944 | . . . . 5
⊢ (𝜑 → ¬ 𝑊 = ∅) | 
| 3 |  | gsumval3a.t | . . . . . . 7
⊢ (𝜑 → 𝑊 ∈ Fin) | 
| 4 |  | fz1f1o 15747 | . . . . . . 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 864 | . . . . 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 495 | . . 3
⊢ (𝜑 → ∃𝑓 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊) | 
| 9 |  | excom 2161 | . . . 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 1860 | . . . . . 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 6918 | . . . . . . 7
⊢ (seq1(
+ ,
(𝐹 ∘ 𝑓))‘(♯‘𝑊)) ∈ V | 
| 12 |  | biidd 262 | . . . . . . 7
⊢ (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) → (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊)) | 
| 13 | 11, 12 | ceqsexv 3531 | . . . . . 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 275 | . . . . 5
⊢
(∃𝑥(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))) ↔ 𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊) | 
| 15 | 14 | exbii 1847 | . . . 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 275 | . . 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 234 | . 2
⊢ (𝜑 → ∃𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))) | 
| 18 |  | exdistrv 1954 | . . . 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 656 | . . . . . 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 480 | . . . . . . . . . 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 18756 | . . . . . . . . . . 11
⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) | 
| 25 | 24 | 3expb 1120 | . . . . . . . . . 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 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | 
| 29 | 28 | sselda 3982 | . . . . . . . . . . 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 717 | . . . . . . . . . 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 772 | . . . . . . . . . 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 19348 | . . . . . . . . . 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 18757 | . . . . . . . . . 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 494 | . . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑊) ∈
ℕ) | 
| 38 | 37 | adantr 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → (♯‘𝑊) ∈
ℕ) | 
| 39 |  | nnuz 12922 | . . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) | 
| 40 | 38, 39 | eleqtrdi 2850 | . . . . . . . . 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 480 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝐹:𝐴⟶𝐵) | 
| 43 | 42 | frnd 6743 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → ran 𝐹 ⊆ 𝐵) | 
| 44 |  | simprr 772 | . . . . . . . . . . 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 6859 | . . . . . . . . . . 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 770 | . . . . . . . . . 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 6870 | . . . . . . . . . 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 6847 | . . . . . . . . . . . 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 7007 | . . . . . . . . . . 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 6736 | . . . . . . . . . . 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 480 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑊 ⊆ 𝐴) | 
| 57 | 51, 56 | fssd 6752 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) → 𝑔:(1...(♯‘𝑊))⟶𝐴) | 
| 58 | 57 | ffvelcdmda 7103 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑥 ∈ (1...(♯‘𝑊))) → (𝑔‘𝑥) ∈ 𝐴) | 
| 59 |  | fnfvelrn 7099 | . . . . . . . . . . 11
⊢ ((𝐹 Fn 𝐴 ∧ (𝑔‘𝑥) ∈ 𝐴) → (𝐹‘(𝑔‘𝑥)) ∈ ran 𝐹) | 
| 60 | 54, 58, 59 | syl2an2r 685 | . . . . . . . . . 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 2840 | . . . . . . . . 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 6847 | . . . . . . . . . . . . . . 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 7007 | . . . . . . . . . . . . . 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 6909 | . . . . . . . . . . . 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 | ffvelcdmda 7103 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) ∧ 𝑘 ∈ (1...(♯‘𝑊))) → (𝑓‘𝑘) ∈ 𝑊) | 
| 68 |  | f1ocnvfv2 7298 | . . . . . . . . . . . . 13
⊢ ((𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ (𝑓‘𝑘) ∈ 𝑊) → (𝑔‘(◡𝑔‘(𝑓‘𝑘))) = (𝑓‘𝑘)) | 
| 69 | 44, 67, 68 | syl2an2r 685 | . . . . . . . . . . . 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 2777 | . . . . . . . . . . 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 6909 | . . . . . . . . . 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 7007 | . . . . . . . . . . 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 6847 | . . . . . . . . . . . . 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 | ffvelcdmda 7103 | . . . . . . . . . . 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 7007 | . . . . . . . . . . 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 685 | . . . . . . . . . 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 2786 | . . . . . . . . 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 14085 | . . . . . . . 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 2753 | . . . . . . . 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 247 | . . . . . . 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 453 | . . . . . 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 | biimtrrid 243 | . . . . 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 1933 | . . . 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 | biimtrrid 243 | . . 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 1927 | . 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 2740 | . . . . . 6
⊢ (𝑥 = 𝑦 → (𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))) | 
| 89 | 88 | anbi2d 630 | . . . . 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 1920 | . . . 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 6835 | . . . . . 6
⊢ (𝑓 = 𝑔 → (𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ↔ 𝑔:(1...(♯‘𝑊))–1-1-onto→𝑊)) | 
| 92 |  | coeq2 5868 | . . . . . . . . 9
⊢ (𝑓 = 𝑔 → (𝐹 ∘ 𝑓) = (𝐹 ∘ 𝑔)) | 
| 93 | 92 | seqeq3d 14051 | . . . . . . . 8
⊢ (𝑓 = 𝑔 → seq1( + , (𝐹 ∘ 𝑓)) = seq1( + , (𝐹 ∘ 𝑔))) | 
| 94 | 93 | fveq1d 6907 | . . . . . . 7
⊢ (𝑓 = 𝑔 → (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊))) | 
| 95 | 94 | eqeq2d 2747 | . . . . . 6
⊢ (𝑓 = 𝑔 → (𝑦 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)) ↔ 𝑦 = (seq1( + , (𝐹 ∘ 𝑔))‘(♯‘𝑊)))) | 
| 96 | 91, 95 | anbi12d 632 | . . . . 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 2035 | . . . 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 2614 | . 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)))) |