MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac12lem1 Structured version   Visualization version   GIF version

Theorem dfac12lem1 9167
Description: Lemma for dfac12 9173. (Contributed by Mario Carneiro, 29-May-2015.)
Hypotheses
Ref Expression
dfac12.1 (𝜑𝐴 ∈ On)
dfac12.3 (𝜑𝐹:𝒫 (har‘(𝑅1𝐴))–1-1→On)
dfac12.4 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))
dfac12.5 (𝜑𝐶 ∈ On)
dfac12.h 𝐻 = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶))
Assertion
Ref Expression
dfac12lem1 (𝜑 → (𝐺𝐶) = (𝑦 ∈ (𝑅1𝐶) ↦ if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦)))))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐶   𝑥,𝐺,𝑦   𝜑,𝑦   𝑥,𝐹,𝑦   𝑦,𝐻
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐻(𝑥)

Proof of Theorem dfac12lem1
StepHypRef Expression
1 dfac12.5 . . 3 (𝜑𝐶 ∈ On)
2 dfac12.4 . . . 4 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))
32tfr2 7647 . . 3 (𝐶 ∈ On → (𝐺𝐶) = ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))))‘(𝐺𝐶)))
41, 3syl 17 . 2 (𝜑 → (𝐺𝐶) = ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))))‘(𝐺𝐶)))
52tfr1 7646 . . . . 5 𝐺 Fn On
6 fnfun 6128 . . . . 5 (𝐺 Fn On → Fun 𝐺)
75, 6ax-mp 5 . . . 4 Fun 𝐺
8 resfunexg 6623 . . . 4 ((Fun 𝐺𝐶 ∈ On) → (𝐺𝐶) ∈ V)
97, 1, 8sylancr 575 . . 3 (𝜑 → (𝐺𝐶) ∈ V)
10 dmeq 5462 . . . . . 6 (𝑥 = (𝐺𝐶) → dom 𝑥 = dom (𝐺𝐶))
1110fveq2d 6336 . . . . 5 (𝑥 = (𝐺𝐶) → (𝑅1‘dom 𝑥) = (𝑅1‘dom (𝐺𝐶)))
1210unieqd 4584 . . . . . . 7 (𝑥 = (𝐺𝐶) → dom 𝑥 = dom (𝐺𝐶))
1310, 12eqeq12d 2786 . . . . . 6 (𝑥 = (𝐺𝐶) → (dom 𝑥 = dom 𝑥 ↔ dom (𝐺𝐶) = dom (𝐺𝐶)))
14 rneq 5489 . . . . . . . . . . . . 13 (𝑥 = (𝐺𝐶) → ran 𝑥 = ran (𝐺𝐶))
15 df-ima 5262 . . . . . . . . . . . . 13 (𝐺𝐶) = ran (𝐺𝐶)
1614, 15syl6eqr 2823 . . . . . . . . . . . 12 (𝑥 = (𝐺𝐶) → ran 𝑥 = (𝐺𝐶))
1716unieqd 4584 . . . . . . . . . . 11 (𝑥 = (𝐺𝐶) → ran 𝑥 = (𝐺𝐶))
1817rneqd 5491 . . . . . . . . . 10 (𝑥 = (𝐺𝐶) → ran ran 𝑥 = ran (𝐺𝐶))
1918unieqd 4584 . . . . . . . . 9 (𝑥 = (𝐺𝐶) → ran ran 𝑥 = ran (𝐺𝐶))
20 suceq 5933 . . . . . . . . 9 ( ran ran 𝑥 = ran (𝐺𝐶) → suc ran ran 𝑥 = suc ran (𝐺𝐶))
2119, 20syl 17 . . . . . . . 8 (𝑥 = (𝐺𝐶) → suc ran ran 𝑥 = suc ran (𝐺𝐶))
2221oveq1d 6808 . . . . . . 7 (𝑥 = (𝐺𝐶) → (suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) = (suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)))
23 fveq1 6331 . . . . . . . 8 (𝑥 = (𝐺𝐶) → (𝑥‘suc (rank‘𝑦)) = ((𝐺𝐶)‘suc (rank‘𝑦)))
2423fveq1d 6334 . . . . . . 7 (𝑥 = (𝐺𝐶) → ((𝑥‘suc (rank‘𝑦))‘𝑦) = (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦))
2522, 24oveq12d 6811 . . . . . 6 (𝑥 = (𝐺𝐶) → ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)) = ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)))
26 id 22 . . . . . . . . . . . . 13 (𝑥 = (𝐺𝐶) → 𝑥 = (𝐺𝐶))
2726, 12fveq12d 6338 . . . . . . . . . . . 12 (𝑥 = (𝐺𝐶) → (𝑥 dom 𝑥) = ((𝐺𝐶)‘ dom (𝐺𝐶)))
2827rneqd 5491 . . . . . . . . . . 11 (𝑥 = (𝐺𝐶) → ran (𝑥 dom 𝑥) = ran ((𝐺𝐶)‘ dom (𝐺𝐶)))
29 oieq2 8574 . . . . . . . . . . 11 (ran (𝑥 dom 𝑥) = ran ((𝐺𝐶)‘ dom (𝐺𝐶)) → OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))))
3028, 29syl 17 . . . . . . . . . 10 (𝑥 = (𝐺𝐶) → OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))))
3130cnveqd 5436 . . . . . . . . 9 (𝑥 = (𝐺𝐶) → OrdIso( E , ran (𝑥 dom 𝑥)) = OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))))
3231, 27coeq12d 5425 . . . . . . . 8 (𝑥 = (𝐺𝐶) → (OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) = (OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))))
3332imaeq1d 5606 . . . . . . 7 (𝑥 = (𝐺𝐶) → ((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦) = ((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))
3433fveq2d 6336 . . . . . 6 (𝑥 = (𝐺𝐶) → (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)) = (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))
3513, 25, 34ifbieq12d 4252 . . . . 5 (𝑥 = (𝐺𝐶) → if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))) = if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))))
3611, 35mpteq12dv 4867 . . . 4 (𝑥 = (𝐺𝐶) → (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))) = (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))))
37 eqid 2771 . . . 4 (𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))) = (𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))))
38 fvex 6342 . . . . 5 (𝑅1‘dom (𝐺𝐶)) ∈ V
3938mptex 6630 . . . 4 (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))) ∈ V
4036, 37, 39fvmpt 6424 . . 3 ((𝐺𝐶) ∈ V → ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))))‘(𝐺𝐶)) = (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))))
419, 40syl 17 . 2 (𝜑 → ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦)))))‘(𝐺𝐶)) = (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))))
42 onss 7137 . . . . . . . 8 (𝐶 ∈ On → 𝐶 ⊆ On)
431, 42syl 17 . . . . . . 7 (𝜑𝐶 ⊆ On)
44 fnssres 6144 . . . . . . 7 ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺𝐶) Fn 𝐶)
455, 43, 44sylancr 575 . . . . . 6 (𝜑 → (𝐺𝐶) Fn 𝐶)
46 fndm 6130 . . . . . 6 ((𝐺𝐶) Fn 𝐶 → dom (𝐺𝐶) = 𝐶)
4745, 46syl 17 . . . . 5 (𝜑 → dom (𝐺𝐶) = 𝐶)
4847fveq2d 6336 . . . 4 (𝜑 → (𝑅1‘dom (𝐺𝐶)) = (𝑅1𝐶))
4948mpteq1d 4872 . . 3 (𝜑 → (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1𝐶) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))))
5047adantr 466 . . . . . . 7 ((𝜑𝑦 ∈ (𝑅1𝐶)) → dom (𝐺𝐶) = 𝐶)
5150unieqd 4584 . . . . . . 7 ((𝜑𝑦 ∈ (𝑅1𝐶)) → dom (𝐺𝐶) = 𝐶)
5250, 51eqeq12d 2786 . . . . . 6 ((𝜑𝑦 ∈ (𝑅1𝐶)) → (dom (𝐺𝐶) = dom (𝐺𝐶) ↔ 𝐶 = 𝐶))
5352ifbid 4247 . . . . 5 ((𝜑𝑦 ∈ (𝑅1𝐶)) → if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))) = if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))))
54 rankr1ai 8825 . . . . . . . . . . . 12 (𝑦 ∈ (𝑅1𝐶) → (rank‘𝑦) ∈ 𝐶)
5554ad2antlr 706 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (rank‘𝑦) ∈ 𝐶)
56 simpr 471 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → 𝐶 = 𝐶)
5755, 56eleqtrd 2852 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (rank‘𝑦) ∈ 𝐶)
58 eloni 5876 . . . . . . . . . . . 12 (𝐶 ∈ On → Ord 𝐶)
59 ordsucuniel 7171 . . . . . . . . . . . 12 (Ord 𝐶 → ((rank‘𝑦) ∈ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
601, 58, 593syl 18 . . . . . . . . . . 11 (𝜑 → ((rank‘𝑦) ∈ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
6160ad2antrr 705 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → ((rank‘𝑦) ∈ 𝐶 ↔ suc (rank‘𝑦) ∈ 𝐶))
6257, 61mpbid 222 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → suc (rank‘𝑦) ∈ 𝐶)
63 fvres 6348 . . . . . . . . 9 (suc (rank‘𝑦) ∈ 𝐶 → ((𝐺𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦)))
6462, 63syl 17 . . . . . . . 8 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → ((𝐺𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦)))
6564fveq1d 6334 . . . . . . 7 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑦))
6665oveq2d 6809 . . . . . 6 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ 𝐶 = 𝐶) → ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)) = ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)))
6766ifeq1da 4255 . . . . 5 ((𝜑𝑦 ∈ (𝑅1𝐶)) → if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))) = if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))))
6851adantr 466 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → dom (𝐺𝐶) = 𝐶)
6968fveq2d 6336 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ dom (𝐺𝐶)) = ((𝐺𝐶)‘ 𝐶))
701ad2antrr 705 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 ∈ On)
71 uniexg 7102 . . . . . . . . . . . . . . . . 17 (𝐶 ∈ On → 𝐶 ∈ V)
72 sucidg 5946 . . . . . . . . . . . . . . . . 17 ( 𝐶 ∈ V → 𝐶 ∈ suc 𝐶)
7370, 71, 723syl 18 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 ∈ suc 𝐶)
741adantr 466 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (𝑅1𝐶)) → 𝐶 ∈ On)
75 orduniorsuc 7177 . . . . . . . . . . . . . . . . . 18 (Ord 𝐶 → (𝐶 = 𝐶𝐶 = suc 𝐶))
7674, 58, 753syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (𝑅1𝐶)) → (𝐶 = 𝐶𝐶 = suc 𝐶))
7776orcanai 977 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶 = suc 𝐶)
7873, 77eleqtrrd 2853 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → 𝐶𝐶)
79 fvres 6348 . . . . . . . . . . . . . . 15 ( 𝐶𝐶 → ((𝐺𝐶)‘ 𝐶) = (𝐺 𝐶))
8078, 79syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ 𝐶) = (𝐺 𝐶))
8169, 80eqtrd 2805 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((𝐺𝐶)‘ dom (𝐺𝐶)) = (𝐺 𝐶))
8281rneqd 5491 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ran ((𝐺𝐶)‘ dom (𝐺𝐶)) = ran (𝐺 𝐶))
83 oieq2 8574 . . . . . . . . . . . 12 (ran ((𝐺𝐶)‘ dom (𝐺𝐶)) = ran (𝐺 𝐶) → OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) = OrdIso( E , ran (𝐺 𝐶)))
8482, 83syl 17 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) = OrdIso( E , ran (𝐺 𝐶)))
8584cnveqd 5436 . . . . . . . . . 10 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) = OrdIso( E , ran (𝐺 𝐶)))
8685, 81coeq12d 5425 . . . . . . . . 9 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → (OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶)))
87 dfac12.h . . . . . . . . 9 𝐻 = (OrdIso( E , ran (𝐺 𝐶)) ∘ (𝐺 𝐶))
8886, 87syl6eqr 2823 . . . . . . . 8 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → (OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) = 𝐻)
8988imaeq1d 5606 . . . . . . 7 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → ((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦) = (𝐻𝑦))
9089fveq2d 6336 . . . . . 6 (((𝜑𝑦 ∈ (𝑅1𝐶)) ∧ ¬ 𝐶 = 𝐶) → (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)) = (𝐹‘(𝐻𝑦)))
9190ifeq2da 4256 . . . . 5 ((𝜑𝑦 ∈ (𝑅1𝐶)) → if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))) = if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦))))
9253, 67, 913eqtrd 2809 . . . 4 ((𝜑𝑦 ∈ (𝑅1𝐶)) → if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦))) = if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦))))
9392mpteq2dva 4878 . . 3 (𝜑 → (𝑦 ∈ (𝑅1𝐶) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1𝐶) ↦ if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦)))))
9449, 93eqtrd 2805 . 2 (𝜑 → (𝑦 ∈ (𝑅1‘dom (𝐺𝐶)) ↦ if(dom (𝐺𝐶) = dom (𝐺𝐶), ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 (((𝐺𝐶)‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran ((𝐺𝐶)‘ dom (𝐺𝐶))) ∘ ((𝐺𝐶)‘ dom (𝐺𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1𝐶) ↦ if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦)))))
954, 41, 943eqtrd 2809 1 (𝜑 → (𝐺𝐶) = (𝑦 ∈ (𝑅1𝐶) ↦ if(𝐶 = 𝐶, ((suc ran (𝐺𝐶) ·𝑜 (rank‘𝑦)) +𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 834   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723  ifcif 4225  𝒫 cpw 4297   cuni 4574  cmpt 4863   E cep 5161  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  ccom 5253  Ord word 5865  Oncon0 5866  suc csuc 5868  Fun wfun 6025   Fn wfn 6026  1-1wf1 6028  cfv 6031  (class class class)co 6793  recscrecs 7620   +𝑜 coa 7710   ·𝑜 comu 7711  OrdIsocoi 8570  harchar 8617  𝑅1cr1 8789  rankcrnk 8790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oi 8571  df-r1 8791  df-rank 8792
This theorem is referenced by:  dfac12lem2  9168
  Copyright terms: Public domain W3C validator