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Theorem oesuclem 7551
Description: Lemma for oesuc 7553. (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 6612 . . . 4 (𝐴 = ∅ → (𝐴𝑜 suc 𝐵) = (∅ ↑𝑜 suc 𝐵))
2 oesuclem.1 . . . . . . . 8 Lim 𝑋
3 limord 5746 . . . . . . . 8 (Lim 𝑋 → Ord 𝑋)
42, 3ax-mp 5 . . . . . . 7 Ord 𝑋
5 ordelord 5707 . . . . . . 7 ((Ord 𝑋𝐵𝑋) → Ord 𝐵)
64, 5mpan 705 . . . . . 6 (𝐵𝑋 → Ord 𝐵)
7 0elsuc 6983 . . . . . 6 (Ord 𝐵 → ∅ ∈ suc 𝐵)
86, 7syl 17 . . . . 5 (𝐵𝑋 → ∅ ∈ suc 𝐵)
9 limsuc 6997 . . . . . . 7 (Lim 𝑋 → (𝐵𝑋 ↔ suc 𝐵𝑋))
102, 9ax-mp 5 . . . . . 6 (𝐵𝑋 ↔ suc 𝐵𝑋)
11 ordelon 5709 . . . . . . . 8 ((Ord 𝑋 ∧ suc 𝐵𝑋) → suc 𝐵 ∈ On)
124, 11mpan 705 . . . . . . 7 (suc 𝐵𝑋 → suc 𝐵 ∈ On)
13 oe0m1 7547 . . . . . . 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 2682 . . 3 ((𝐵𝑋𝐴 = ∅) → (𝐴𝑜 suc 𝐵) = ∅)
18 oveq1 6612 . . . . 5 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
19 id 22 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
2018, 19oveq12d 6623 . . . 4 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ·𝑜 𝐴) = ((∅ ↑𝑜 𝐵) ·𝑜 ∅))
21 ordelon 5709 . . . . . . 7 ((Ord 𝑋𝐵𝑋) → 𝐵 ∈ On)
224, 21mpan 705 . . . . . 6 (𝐵𝑋𝐵 ∈ On)
23 oveq2 6613 . . . . . . . . 9 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
24 oe0m0 7546 . . . . . . . . . 10 (∅ ↑𝑜 ∅) = 1𝑜
25 1on 7513 . . . . . . . . . 10 1𝑜 ∈ On
2624, 25eqeltri 2700 . . . . . . . . 9 (∅ ↑𝑜 ∅) ∈ On
2723, 26syl6eqel 2712 . . . . . . . 8 (𝐵 = ∅ → (∅ ↑𝑜 𝐵) ∈ On)
2827adantl 482 . . . . . . 7 ((𝐵𝑋𝐵 = ∅) → (∅ ↑𝑜 𝐵) ∈ On)
29 oe0m1 7547 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
3022, 29syl 17 . . . . . . . . . 10 (𝐵𝑋 → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
3130biimpa 501 . . . . . . . . 9 ((𝐵𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
32 0elon 5740 . . . . . . . . 9 ∅ ∈ On
3331, 32syl6eqel 2712 . . . . . . . 8 ((𝐵𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
3433adantll 749 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
3528, 34oe0lem 7539 . . . . . 6 ((𝐵 ∈ On ∧ 𝐵𝑋) → (∅ ↑𝑜 𝐵) ∈ On)
3622, 35mpancom 702 . . . . 5 (𝐵𝑋 → (∅ ↑𝑜 𝐵) ∈ On)
37 om0 7543 . . . . 5 ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
3836, 37syl 17 . . . 4 (𝐵𝑋 → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
3920, 38sylan9eqr 2682 . . 3 ((𝐵𝑋𝐴 = ∅) → ((𝐴𝑜 𝐵) ·𝑜 𝐴) = ∅)
4017, 39eqtr4d 2663 . 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 762 . . 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 7541 . . . 4 (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵))
4543, 44sylanl2 682 . . 3 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵))
46 ovex 6633 . . . . 5 (𝐴𝑜 𝐵) ∈ V
47 oveq1 6612 . . . . . 6 (𝑥 = (𝐴𝑜 𝐵) → (𝑥 ·𝑜 𝐴) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))
48 eqid 2626 . . . . . 6 (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))
49 ovex 6633 . . . . . 6 ((𝐴𝑜 𝐵) ·𝑜 𝐴) ∈ V
5047, 48, 49fvmpt 6240 . . . . 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 7541 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
5322, 52sylanl2 682 . . . . 5 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
5453fveq2d 6154 . . . 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 2674 . . 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 2670 . 2 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 suc 𝐵) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))
5740, 56oe0lem 7539 1 ((𝐴 ∈ On ∧ 𝐵𝑋) → (𝐴𝑜 suc 𝐵) = ((𝐴𝑜 𝐵) ·𝑜 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  Vcvv 3191  c0 3896  cmpt 4678  Ord word 5684  Oncon0 5685  Lim wlim 5686  suc csuc 5687  cfv 5850  (class class class)co 6605  reccrdg 7451  1𝑜c1o 7499   ·𝑜 comu 7504  𝑜 coe 7505
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-omul 7511  df-oexp 7512
This theorem is referenced by:  oesuc  7553  onesuc  7556
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