Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  finxpreclem4 Structured version   Visualization version   GIF version

Theorem finxpreclem4 36275
Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.)
Hypothesis
Ref Expression
finxpreclem4.1 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
Assertion
Ref Expression
finxpreclem4 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁))
Distinct variable groups:   𝑛,𝑁,𝑥   𝑈,𝑛,𝑥   𝑦,𝑛,𝑥
Allowed substitution hints:   𝑈(𝑦)   𝐹(𝑥,𝑦,𝑛)   𝑁(𝑦)

Proof of Theorem finxpreclem4
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 2onn 8641 . . . . . . . 8 2o ∈ ω
2 nnon 7861 . . . . . . . . . . 11 (𝑁 ∈ ω → 𝑁 ∈ On)
3 2on 8480 . . . . . . . . . . . . . 14 2o ∈ On
4 oawordeu 8555 . . . . . . . . . . . . . 14 (((2o ∈ On ∧ 𝑁 ∈ On) ∧ 2o𝑁) → ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁)
53, 4mpanl1 699 . . . . . . . . . . . . 13 ((𝑁 ∈ On ∧ 2o𝑁) → ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁)
6 riotasbc 7384 . . . . . . . . . . . . 13 (∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁[(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁)
75, 6syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ On ∧ 2o𝑁) → [(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁)
8 riotaex 7369 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ V
9 sbceq1g 4415 . . . . . . . . . . . . . 14 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ V → ([(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = 𝑁))
108, 9ax-mp 5 . . . . . . . . . . . . 13 ([(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = 𝑁)
11 csbov2g 7455 . . . . . . . . . . . . . . . 16 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ V → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜𝑜))
128, 11ax-mp 5 . . . . . . . . . . . . . . 15 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜𝑜)
138csbvargi 4433 . . . . . . . . . . . . . . . 16 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜𝑜 = (𝑜 ∈ On (2o +o 𝑜) = 𝑁)
1413oveq2i 7420 . . . . . . . . . . . . . . 15 (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜𝑜) = (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))
1512, 14eqtri 2761 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))
1615eqeq1i 2738 . . . . . . . . . . . . 13 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜(2o +o 𝑜) = 𝑁 ↔ (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
1710, 16bitri 275 . . . . . . . . . . . 12 ([(𝑜 ∈ On (2o +o 𝑜) = 𝑁) / 𝑜](2o +o 𝑜) = 𝑁 ↔ (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
187, 17sylib 217 . . . . . . . . . . 11 ((𝑁 ∈ On ∧ 2o𝑁) → (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
192, 18sylan 581 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2o𝑁) → (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = 𝑁)
20 simpl 484 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 ∈ ω)
2119, 20eqeltrd 2834 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 2o𝑁) → (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω)
22 riotacl 7383 . . . . . . . . . . 11 (∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On)
23 riotaund 7405 . . . . . . . . . . . 12 (¬ ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) = ∅)
24 0elon 6419 . . . . . . . . . . . 12 ∅ ∈ On
2523, 24eqeltrdi 2842 . . . . . . . . . . 11 (¬ ∃!𝑜 ∈ On (2o +o 𝑜) = 𝑁 → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On)
2622, 25pm2.61i 182 . . . . . . . . . 10 (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On
27 nnarcl 8616 . . . . . . . . . . . 12 ((2o ∈ On ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On) → ((2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω ↔ (2o ∈ ω ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)))
283, 27mpan 689 . . . . . . . . . . 11 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On → ((2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω ↔ (2o ∈ ω ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)))
291biantrur 532 . . . . . . . . . . 11 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω ↔ (2o ∈ ω ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω))
3028, 29bitr4di 289 . . . . . . . . . 10 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ On → ((2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω ↔ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω))
3126, 30ax-mp 5 . . . . . . . . 9 ((2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω ↔ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)
3221, 31sylib 217 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2o𝑁) → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)
33 nnacom 8617 . . . . . . . 8 ((2o ∈ ω ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω) → (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o))
341, 32, 33sylancr 588 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → (2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) = ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o))
35 df-2o 8467 . . . . . . . . 9 2o = suc 1o
3635oveq2i 7420 . . . . . . . 8 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o) = ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o suc 1o)
37 1onn 8639 . . . . . . . . 9 1o ∈ ω
38 nnasuc 8606 . . . . . . . . 9 (((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω ∧ 1o ∈ ω) → ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o suc 1o) = suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o))
3932, 37, 38sylancl 587 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2o𝑁) → ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o suc 1o) = suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o))
4036, 39eqtrid 2785 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 2o) = suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o))
4134, 19, 403eqtr3d 2781 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 = suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o))
422adantr 482 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 ∈ On)
43 sucidg 6446 . . . . . . . . . . . 12 (1o ∈ ω → 1o ∈ suc 1o)
4437, 43ax-mp 5 . . . . . . . . . . 11 1o ∈ suc 1o
4544, 35eleqtrri 2833 . . . . . . . . . 10 1o ∈ 2o
46 ssel 3976 . . . . . . . . . 10 (2o𝑁 → (1o ∈ 2o → 1o𝑁))
4745, 46mpi 20 . . . . . . . . 9 (2o𝑁 → 1o𝑁)
4847ne0d 4336 . . . . . . . 8 (2o𝑁𝑁 ≠ ∅)
4948adantl 483 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 ≠ ∅)
50 nnlim 7869 . . . . . . . 8 (𝑁 ∈ ω → ¬ Lim 𝑁)
5150adantr 482 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → ¬ Lim 𝑁)
52 onsucuni3 36248 . . . . . . 7 ((𝑁 ∈ On ∧ 𝑁 ≠ ∅ ∧ ¬ Lim 𝑁) → 𝑁 = suc 𝑁)
5342, 49, 51, 52syl3anc 1372 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 = suc 𝑁)
54 nnacom 8617 . . . . . . . 8 (((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω ∧ 1o ∈ ω) → ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o) = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
5532, 37, 54sylancl 587 . . . . . . 7 ((𝑁 ∈ ω ∧ 2o𝑁) → ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o) = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
56 suceq 6431 . . . . . . 7 (((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o) = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) → suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o) = suc (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
5755, 56syl 17 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → suc ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) +o 1o) = suc (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
5841, 53, 573eqtr3d 2781 . . . . 5 ((𝑁 ∈ ω ∧ 2o𝑁) → suc 𝑁 = suc (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
59 ordom 7865 . . . . . . . . 9 Ord ω
60 ordelss 6381 . . . . . . . . 9 ((Ord ω ∧ 𝑁 ∈ ω) → 𝑁 ⊆ ω)
6159, 60mpan 689 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ⊆ ω)
62 nnfi 9167 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ∈ Fin)
63 nnunifi 9294 . . . . . . . 8 ((𝑁 ⊆ ω ∧ 𝑁 ∈ Fin) → 𝑁 ∈ ω)
6461, 62, 63syl2anc 585 . . . . . . 7 (𝑁 ∈ ω → 𝑁 ∈ ω)
6564adantr 482 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 ∈ ω)
66 nnacl 8611 . . . . . . 7 ((1o ∈ ω ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω) → (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω)
6737, 32, 66sylancr 588 . . . . . 6 ((𝑁 ∈ ω ∧ 2o𝑁) → (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω)
68 peano4 7883 . . . . . 6 (( 𝑁 ∈ ω ∧ (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ∈ ω) → (suc 𝑁 = suc (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ↔ 𝑁 = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))))
6965, 67, 68syl2anc 585 . . . . 5 ((𝑁 ∈ ω ∧ 2o𝑁) → (suc 𝑁 = suc (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)) ↔ 𝑁 = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))))
7058, 69mpbid 231 . . . 4 ((𝑁 ∈ ω ∧ 2o𝑁) → 𝑁 = (1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))
7170fveq2d 6896 . . 3 ((𝑁 ∈ ω ∧ 2o𝑁) → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))))
7271adantr 482 . 2 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))))
7332adantr 482 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω)
74 df-1o 8466 . . . . . . . 8 1o = suc ∅
7574fveq2i 6895 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅)
76 rdgsuc 8424 . . . . . . . 8 (∅ ∈ On → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)))
7724, 76ax-mp 5 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅))
78 opex 5465 . . . . . . . . 9 𝑁, 𝑦⟩ ∈ V
7978rdg0 8421 . . . . . . . 8 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅) = ⟨𝑁, 𝑦
8079fveq2i 6895 . . . . . . 7 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)) = (𝐹‘⟨𝑁, 𝑦⟩)
8175, 77, 803eqtri 2765 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (𝐹‘⟨𝑁, 𝑦⟩)
82 finxpreclem4.1 . . . . . . 7 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
8382finxpreclem3 36274 . . . . . 6 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑦)⟩ = (𝐹‘⟨𝑁, 𝑦⟩))
8481, 83eqtr4id 2792 . . . . 5 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = ⟨ 𝑁, (1st𝑦)⟩)
8584fveq2d 6896 . . . 4 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩))
86 2on0 8482 . . . . . 6 2o ≠ ∅
87 nnlim 7869 . . . . . . 7 (2o ∈ ω → ¬ Lim 2o)
881, 87ax-mp 5 . . . . . 6 ¬ Lim 2o
89 rdgsucuni 36250 . . . . . 6 ((2o ∈ On ∧ 2o ≠ ∅ ∧ ¬ Lim 2o) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o)))
903, 86, 88, 89mp3an 1462 . . . . 5 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o))
91 1oequni2o 36249 . . . . . . 7 1o = 2o
9291fveq2i 6895 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o)
9392fveq2i 6895 . . . . 5 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o)) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2o))
9490, 93eqtr4i 2764 . . . 4 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1o))
9574fveq2i 6895 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅)
96 rdgsuc 8424 . . . . . 6 (∅ ∈ On → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)))
9724, 96ax-mp 5 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅))
98 opex 5465 . . . . . . 7 𝑁, (1st𝑦)⟩ ∈ V
9998rdg0 8421 . . . . . 6 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅) = ⟨ 𝑁, (1st𝑦)⟩
10099fveq2i 6895 . . . . 5 (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10195, 97, 1003eqtri 2765 . . . 4 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10285, 94, 1013eqtr4g 2798 . . 3 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o))
103 1on 8478 . . . 4 1o ∈ On
104 rdgeqoa 36251 . . . 4 ((2o ∈ On ∧ 1o ∈ On ∧ (𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω) → ((rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))))
1053, 103, 104mp3an12 1452 . . 3 ((𝑜 ∈ On (2o +o 𝑜) = 𝑁) ∈ ω → ((rec(𝐹, ⟨𝑁, 𝑦⟩)‘2o) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1o) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁)))))
10673, 102, 105sylc 65 . 2 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))))
10719fveq2d 6896 . . 3 ((𝑁 ∈ ω ∧ 2o𝑁) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
108107adantr 482 . 2 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2o +o (𝑜 ∈ On (2o +o 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
10972, 106, 1083eqtr2rd 2780 1 (((𝑁 ∈ ω ∧ 2o𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2941  ∃!wreu 3375  Vcvv 3475  [wsbc 3778  csb 3894  wss 3949  c0 4323  ifcif 4529  cop 4635   cuni 4909   × cxp 5675  Ord word 6364  Oncon0 6365  Lim wlim 6366  suc csuc 6367  cfv 6544  crio 7364  (class class class)co 7409  cmpo 7411  ωcom 7855  1st c1st 7973  reccrdg 8409  1oc1o 8459  2oc2o 8460   +o coa 8463  Fincfn 8939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-en 8940  df-fin 8943
This theorem is referenced by:  finxpsuclem  36278
  Copyright terms: Public domain W3C validator