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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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