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 32863
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 7665 . . . . . . . 8 2𝑜 ∈ ω
2 nnon 7018 . . . . . . . . . . 11 (𝑁 ∈ ω → 𝑁 ∈ On)
3 2on 7513 . . . . . . . . . . . . . 14 2𝑜 ∈ On
4 oawordeu 7580 . . . . . . . . . . . . . 14 (((2𝑜 ∈ On ∧ 𝑁 ∈ On) ∧ 2𝑜𝑁) → ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
53, 4mpanl1 715 . . . . . . . . . . . . 13 ((𝑁 ∈ On ∧ 2𝑜𝑁) → ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
6 riotasbc 6580 . . . . . . . . . . . . 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 6569 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ V
9 sbceq1g 3960 . . . . . . . . . . . . . 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 6644 . . . . . . . . . . . . . . . 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)
13 csbvarg 3975 . . . . . . . . . . . . . . . . 17 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ V → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜𝑜 = (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
148, 13ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜𝑜 = (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)
1514oveq2i 6615 . . . . . . . . . . . . . . 15 (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜𝑜) = (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
1612, 15eqtri 2643 . . . . . . . . . . . . . 14 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜(2𝑜 +𝑜 𝑜) = (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))
1716eqeq1i 2626 . . . . . . . . . . . . 13 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜(2𝑜 +𝑜 𝑜) = 𝑁 ↔ (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
1810, 17bitri 264 . . . . . . . . . . . 12 ([(𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) / 𝑜](2𝑜 +𝑜 𝑜) = 𝑁 ↔ (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
197, 18sylib 208 . . . . . . . . . . 11 ((𝑁 ∈ On ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
202, 19sylan 488 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = 𝑁)
21 simpl 473 . . . . . . . . . 10 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ ω)
2220, 21eqeltrd 2698 . . . . . . . . 9 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
23 riotacl 6579 . . . . . . . . . . 11 (∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On)
24 riotaund 6601 . . . . . . . . . . . 12 (¬ ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) = ∅)
25 0elon 5737 . . . . . . . . . . . 12 ∅ ∈ On
2624, 25syl6eqel 2706 . . . . . . . . . . 11 (¬ ∃!𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁 → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On)
2723, 26pm2.61i 176 . . . . . . . . . 10 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On
28 nnarcl 7641 . . . . . . . . . . . 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)))
293, 28mpan 705 . . . . . . . . . . 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)))
301biantrur 527 . . . . . . . . . . 11 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω ↔ (2𝑜 ∈ ω ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω))
3129, 30syl6bbr 278 . . . . . . . . . 10 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ On → ((2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω ↔ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω))
3227, 31ax-mp 5 . . . . . . . . 9 ((2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω ↔ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
3322, 32sylib 208 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
34 nnacom 7642 . . . . . . . 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))
351, 33, 34sylancr 694 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) = ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜))
36 df-2o 7506 . . . . . . . . 9 2𝑜 = suc 1𝑜
3736oveq2i 6615 . . . . . . . 8 ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜) = ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜)
38 1onn 7664 . . . . . . . . 9 1𝑜 ∈ ω
39 nnasuc 7631 . . . . . . . . 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))
4033, 38, 39sylancl 693 . . . . . . . 8 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 suc 1𝑜) = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
4137, 40syl5eq 2667 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 2𝑜) = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
4235, 20, 413eqtr3d 2663 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 = suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜))
432adantr 481 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ On)
44 sucidg 5762 . . . . . . . . . . . 12 (1𝑜 ∈ ω → 1𝑜 ∈ suc 1𝑜)
4538, 44ax-mp 5 . . . . . . . . . . 11 1𝑜 ∈ suc 1𝑜
4645, 36eleqtrri 2697 . . . . . . . . . 10 1𝑜 ∈ 2𝑜
47 ssel 3577 . . . . . . . . . 10 (2𝑜𝑁 → (1𝑜 ∈ 2𝑜 → 1𝑜𝑁))
4846, 47mpi 20 . . . . . . . . 9 (2𝑜𝑁 → 1𝑜𝑁)
49 ne0i 3897 . . . . . . . . 9 (1𝑜𝑁𝑁 ≠ ∅)
5048, 49syl 17 . . . . . . . 8 (2𝑜𝑁𝑁 ≠ ∅)
5150adantl 482 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ≠ ∅)
52 nnlim 7025 . . . . . . . 8 (𝑁 ∈ ω → ¬ Lim 𝑁)
5352adantr 481 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ¬ Lim 𝑁)
54 onsucuni3 32847 . . . . . . 7 ((𝑁 ∈ On ∧ 𝑁 ≠ ∅ ∧ ¬ Lim 𝑁) → 𝑁 = suc 𝑁)
5543, 51, 53, 54syl3anc 1323 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 = suc 𝑁)
56 nnacom 7642 . . . . . . . 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)))
5733, 38, 56sylancl 693 . . . . . . 7 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
58 suceq 5749 . . . . . . 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)))
5957, 58syl 17 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → suc ((𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) +𝑜 1𝑜) = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
6042, 55, 593eqtr3d 2663 . . . . 5 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → suc 𝑁 = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
61 ordom 7021 . . . . . . . . 9 Ord ω
62 ordelss 5698 . . . . . . . . 9 ((Ord ω ∧ 𝑁 ∈ ω) → 𝑁 ⊆ ω)
6361, 62mpan 705 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ⊆ ω)
64 nnfi 8097 . . . . . . . 8 (𝑁 ∈ ω → 𝑁 ∈ Fin)
65 nnunifi 8155 . . . . . . . 8 ((𝑁 ⊆ ω ∧ 𝑁 ∈ Fin) → 𝑁 ∈ ω)
6663, 64, 65syl2anc 692 . . . . . . 7 (𝑁 ∈ ω → 𝑁 ∈ ω)
6766adantr 481 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 ∈ ω)
68 nnacl 7636 . . . . . . 7 ((1𝑜 ∈ ω ∧ (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω) → (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
6938, 33, 68sylancr 694 . . . . . 6 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ∈ ω)
70 peano4 7035 . . . . . 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))))
7167, 69, 70syl2anc 692 . . . . 5 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (suc 𝑁 = suc (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)) ↔ 𝑁 = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
7260, 71mpbid 222 . . . 4 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → 𝑁 = (1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁)))
7372fveq2d 6152 . . 3 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
7473adantr 481 . 2 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘ 𝑁) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘(1𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))))
7533adantr 481 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁) ∈ ω)
76 finxpreclem4.1 . . . . . . 7 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
7776finxpreclem3 32862 . . . . . 6 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → ⟨ 𝑁, (1st𝑦)⟩ = (𝐹‘⟨𝑁, 𝑦⟩))
78 df-1o 7505 . . . . . . . 8 1𝑜 = suc ∅
7978fveq2i 6151 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅)
80 rdgsuc 7465 . . . . . . . 8 (∅ ∈ On → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)))
8125, 80ax-mp 5 . . . . . . 7 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅))
82 opex 4893 . . . . . . . . 9 𝑁, 𝑦⟩ ∈ V
8382rdg0 7462 . . . . . . . 8 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅) = ⟨𝑁, 𝑦
8483fveq2i 6151 . . . . . . 7 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘∅)) = (𝐹‘⟨𝑁, 𝑦⟩)
8579, 81, 843eqtri 2647 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (𝐹‘⟨𝑁, 𝑦⟩)
8677, 85syl6reqr 2674 . . . . 5 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = ⟨ 𝑁, (1st𝑦)⟩)
8786fveq2d 6152 . . . 4 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩))
88 2on0 7514 . . . . . 6 2𝑜 ≠ ∅
89 nnlim 7025 . . . . . . 7 (2𝑜 ∈ ω → ¬ Lim 2𝑜)
901, 89ax-mp 5 . . . . . 6 ¬ Lim 2𝑜
91 rdgsucuni 32849 . . . . . 6 ((2𝑜 ∈ On ∧ 2𝑜 ≠ ∅ ∧ ¬ Lim 2𝑜) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜)))
923, 88, 90, 91mp3an 1421 . . . . 5 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜))
93 1oequni2o 32848 . . . . . . 7 1𝑜 = 2𝑜
9493fveq2i 6151 . . . . . 6 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜)
9594fveq2i 6151 . . . . 5 (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜)) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘ 2𝑜))
9692, 95eqtr4i 2646 . . . 4 (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (𝐹‘(rec(𝐹, ⟨𝑁, 𝑦⟩)‘1𝑜))
9778fveq2i 6151 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅)
98 rdgsuc 7465 . . . . . 6 (∅ ∈ On → (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)))
9925, 98ax-mp 5 . . . . 5 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘suc ∅) = (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅))
100 opex 4893 . . . . . . 7 𝑁, (1st𝑦)⟩ ∈ V
101100rdg0 7462 . . . . . 6 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅) = ⟨ 𝑁, (1st𝑦)⟩
102101fveq2i 6151 . . . . 5 (𝐹‘(rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘∅)) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10397, 99, 1023eqtri 2647 . . . 4 (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜) = (𝐹‘⟨ 𝑁, (1st𝑦)⟩)
10487, 96, 1033eqtr4g 2680 . . 3 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘2𝑜) = (rec(𝐹, ⟨ 𝑁, (1st𝑦)⟩)‘1𝑜))
105 1on 7512 . . . 4 1𝑜 ∈ On
106 rdgeqoa 32850 . . . 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)))))
1073, 105, 106mp3an12 1411 . . 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)))))
10875, 104, 107sylc 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))))
10920fveq2d 6152 . . 3 ((𝑁 ∈ ω ∧ 2𝑜𝑁) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
110109adantr 481 . 2 (((𝑁 ∈ ω ∧ 2𝑜𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘(2𝑜 +𝑜 (𝑜 ∈ On (2𝑜 +𝑜 𝑜) = 𝑁))) = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))
11174, 108, 1103eqtr2rd 2662 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 196  wa 384   = wceq 1480  wcel 1987  wne 2790  ∃!wreu 2909  Vcvv 3186  [wsbc 3417  csb 3514  wss 3555  c0 3891  ifcif 4058  cop 4154   cuni 4402   × cxp 5072  Ord word 5681  Oncon0 5682  Lim wlim 5683  suc csuc 5684  cfv 5847  crio 6564  (class class class)co 6604  cmpt2 6606  ωcom 7012  1st c1st 7111  reccrdg 7450  1𝑜c1o 7498  2𝑜c2o 7499   +𝑜 coa 7502  Fincfn 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903
This theorem is referenced by:  finxpsuclem  32866
  Copyright terms: Public domain W3C validator