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Theorem oesuclem 7650
Description: Lemma for oesuc 7652. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
oesuclem.1 Lim 𝑋
oesuclem.2 (𝐵𝑋 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))
Assertion
Ref Expression
oesuclem ((𝐴 ∈ On ∧ 𝐵𝑋) → (𝐴𝑜 suc 𝐵) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem oesuclem
StepHypRef Expression
1 oveq1 6697 . . . 4 (𝐴 = ∅ → (𝐴𝑜 suc 𝐵) = (∅ ↑𝑜 suc 𝐵))
2 oesuclem.1 . . . . . . . 8 Lim 𝑋
3 limord 5822 . . . . . . . 8 (Lim 𝑋 → Ord 𝑋)
42, 3ax-mp 5 . . . . . . 7 Ord 𝑋
5 ordelord 5783 . . . . . . 7 ((Ord 𝑋𝐵𝑋) → Ord 𝐵)
64, 5mpan 706 . . . . . 6 (𝐵𝑋 → Ord 𝐵)
7 0elsuc 7077 . . . . . 6 (Ord 𝐵 → ∅ ∈ suc 𝐵)
86, 7syl 17 . . . . 5 (𝐵𝑋 → ∅ ∈ suc 𝐵)
9 limsuc 7091 . . . . . . 7 (Lim 𝑋 → (𝐵𝑋 ↔ suc 𝐵𝑋))
102, 9ax-mp 5 . . . . . 6 (𝐵𝑋 ↔ suc 𝐵𝑋)
11 ordelon 5785 . . . . . . . 8 ((Ord 𝑋 ∧ suc 𝐵𝑋) → suc 𝐵 ∈ On)
124, 11mpan 706 . . . . . . 7 (suc 𝐵𝑋 → suc 𝐵 ∈ On)
13 oe0m1 7646 . . . . . . 7 (suc 𝐵 ∈ On → (∅ ∈ suc 𝐵 ↔ (∅ ↑𝑜 suc 𝐵) = ∅))
1412, 13syl 17 . . . . . 6 (suc 𝐵𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑𝑜 suc 𝐵) = ∅))
1510, 14sylbi 207 . . . . 5 (𝐵𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑𝑜 suc 𝐵) = ∅))
168, 15mpbid 222 . . . 4 (𝐵𝑋 → (∅ ↑𝑜 suc 𝐵) = ∅)
171, 16sylan9eqr 2707 . . 3 ((𝐵𝑋𝐴 = ∅) → (𝐴𝑜 suc 𝐵) = ∅)
18 oveq1 6697 . . . . 5 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
19 id 22 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
2018, 19oveq12d 6708 . . . 4 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ·𝑜 𝐴) = ((∅ ↑𝑜 𝐵) ·𝑜 ∅))
21 ordelon 5785 . . . . . . 7 ((Ord 𝑋𝐵𝑋) → 𝐵 ∈ On)
224, 21mpan 706 . . . . . 6 (𝐵𝑋𝐵 ∈ On)
23 oveq2 6698 . . . . . . . . 9 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
24 oe0m0 7645 . . . . . . . . . 10 (∅ ↑𝑜 ∅) = 1𝑜
25 1on 7612 . . . . . . . . . 10 1𝑜 ∈ On
2624, 25eqeltri 2726 . . . . . . . . 9 (∅ ↑𝑜 ∅) ∈ On
2723, 26syl6eqel 2738 . . . . . . . 8 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) ∈ On)
2827adantl 481 . . . . . . 7 ((𝐵𝑋𝐵 = ∅) → (∅ ↑𝑜 𝐵) ∈ On)
29 oe0m1 7646 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
3022, 29syl 17 . . . . . . . . . 10 (𝐵𝑋 → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
3130biimpa 500 . . . . . . . . 9 ((𝐵𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
32 0elon 5816 . . . . . . . . 9 ∅ ∈ On
3331, 32syl6eqel 2738 . . . . . . . 8 ((𝐵𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
3433adantll 750 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
3528, 34oe0lem 7638 . . . . . 6 ((𝐵 ∈ On ∧ 𝐵𝑋) → (∅ ↑𝑜 𝐵) ∈ On)
3622, 35mpancom 704 . . . . 5 (𝐵𝑋 → (∅ ↑𝑜 𝐵) ∈ On)
37 om0 7642 . . . . 5 ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
3836, 37syl 17 . . . 4 (𝐵𝑋 → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
3920, 38sylan9eqr 2707 . . 3 ((𝐵𝑋𝐴 = ∅) → ((𝐴𝑜 𝐵) ·𝑜 𝐴) = ∅)
4017, 39eqtr4d 2688 . 2 ((𝐵𝑋𝐴 = ∅) → (𝐴𝑜 suc 𝐵) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))
41 oesuclem.2 . . . 4 (𝐵𝑋 → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))
4241ad2antlr 763 . . 3 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))
4310, 12sylbi 207 . . . 4 (𝐵𝑋 → suc 𝐵 ∈ On)
44 oevn0 7640 . . . 4 (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵))
4543, 44sylanl2 684 . . 3 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵))
46 ovex 6718 . . . . 5 (𝐴𝑜 𝐵) ∈ V
47 oveq1 6697 . . . . . 6 (𝑥 = (𝐴𝑜 𝐵) → (𝑥 ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))
48 eqid 2651 . . . . . 6 (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))
49 ovex 6718 . . . . . 6 ((𝐴𝑜 𝐵) ·𝑜 𝐴) ∈ V
5047, 48, 49fvmpt 6321 . . . . 5 ((𝐴𝑜 𝐵) ∈ V → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘(𝐴𝑜 𝐵)) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))
5146, 50ax-mp 5 . . . 4 ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘(𝐴𝑜 𝐵)) = ((𝐴𝑜 𝐵) ·𝑜 𝐴)
52 oevn0 7640 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
5322, 52sylanl2 684 . . . . 5 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
5453fveq2d 6233 . . . 4 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘(𝐴𝑜 𝐵)) = ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))
5551, 54syl5eqr 2699 . . 3 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → ((𝐴𝑜 𝐵) ·𝑜 𝐴) = ((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)))
5642, 45, 553eqtr4d 2695 . 2 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 suc 𝐵) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))
5740, 56oe0lem 7638 1 ((𝐴 ∈ On ∧ 𝐵𝑋) → (𝐴𝑜 suc 𝐵) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948  cmpt 4762  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  cfv 5926  (class class class)co 6690  reccrdg 7550  1𝑜c1o 7598   ·𝑜 comu 7603  𝑜 coe 7604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-omul 7610  df-oexp 7611
This theorem is referenced by:  oesuc  7652  onesuc  7655
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