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Theorem oesuclem 8144
 Description: Lemma for oesuc 8146. (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 7157 . . . 4 (𝐴 = ∅ → (𝐴o suc 𝐵) = (∅ ↑o suc 𝐵))
2 oesuclem.1 . . . . . . . 8 Lim 𝑋
3 limord 6245 . . . . . . . 8 (Lim 𝑋 → Ord 𝑋)
42, 3ax-mp 5 . . . . . . 7 Ord 𝑋
5 ordelord 6208 . . . . . . 7 ((Ord 𝑋𝐵𝑋) → Ord 𝐵)
64, 5mpan 688 . . . . . 6 (𝐵𝑋 → Ord 𝐵)
7 0elsuc 7544 . . . . . 6 (Ord 𝐵 → ∅ ∈ suc 𝐵)
86, 7syl 17 . . . . 5 (𝐵𝑋 → ∅ ∈ suc 𝐵)
9 limsuc 7558 . . . . . . 7 (Lim 𝑋 → (𝐵𝑋 ↔ suc 𝐵𝑋))
102, 9ax-mp 5 . . . . . 6 (𝐵𝑋 ↔ suc 𝐵𝑋)
11 ordelon 6210 . . . . . . . 8 ((Ord 𝑋 ∧ suc 𝐵𝑋) → suc 𝐵 ∈ On)
124, 11mpan 688 . . . . . . 7 (suc 𝐵𝑋 → suc 𝐵 ∈ On)
13 oe0m1 8140 . . . . . . 7 (suc 𝐵 ∈ On → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
1412, 13syl 17 . . . . . 6 (suc 𝐵𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
1510, 14sylbi 219 . . . . 5 (𝐵𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
168, 15mpbid 234 . . . 4 (𝐵𝑋 → (∅ ↑o suc 𝐵) = ∅)
171, 16sylan9eqr 2878 . . 3 ((𝐵𝑋𝐴 = ∅) → (𝐴o suc 𝐵) = ∅)
18 oveq1 7157 . . . . 5 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
19 id 22 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
2018, 19oveq12d 7168 . . . 4 (𝐴 = ∅ → ((𝐴o 𝐵) ·o 𝐴) = ((∅ ↑o 𝐵) ·o ∅))
21 ordelon 6210 . . . . . . 7 ((Ord 𝑋𝐵𝑋) → 𝐵 ∈ On)
224, 21mpan 688 . . . . . 6 (𝐵𝑋𝐵 ∈ On)
23 oveq2 7158 . . . . . . . . 9 (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
24 oe0m0 8139 . . . . . . . . . 10 (∅ ↑o ∅) = 1o
25 1on 8103 . . . . . . . . . 10 1o ∈ On
2624, 25eqeltri 2909 . . . . . . . . 9 (∅ ↑o ∅) ∈ On
2723, 26eqeltrdi 2921 . . . . . . . 8 (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
2827adantl 484 . . . . . . 7 ((𝐵𝑋𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
29 oe0m1 8140 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
3022, 29syl 17 . . . . . . . . . 10 (𝐵𝑋 → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
3130biimpa 479 . . . . . . . . 9 ((𝐵𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
32 0elon 6239 . . . . . . . . 9 ∅ ∈ On
3331, 32eqeltrdi 2921 . . . . . . . 8 ((𝐵𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
3433adantll 712 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
3528, 34oe0lem 8132 . . . . . 6 ((𝐵 ∈ On ∧ 𝐵𝑋) → (∅ ↑o 𝐵) ∈ On)
3622, 35mpancom 686 . . . . 5 (𝐵𝑋 → (∅ ↑o 𝐵) ∈ On)
37 om0 8136 . . . . 5 ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ·o ∅) = ∅)
3836, 37syl 17 . . . 4 (𝐵𝑋 → ((∅ ↑o 𝐵) ·o ∅) = ∅)
3920, 38sylan9eqr 2878 . . 3 ((𝐵𝑋𝐴 = ∅) → ((𝐴o 𝐵) ·o 𝐴) = ∅)
4017, 39eqtr4d 2859 . 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 725 . . 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 219 . . . 4 (𝐵𝑋 → suc 𝐵 ∈ On)
44 oevn0 8134 . . . 4 (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
4543, 44sylanl2 679 . . 3 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
46 ovex 7183 . . . . 5 (𝐴o 𝐵) ∈ V
47 oveq1 7157 . . . . . 6 (𝑥 = (𝐴o 𝐵) → (𝑥 ·o 𝐴) = ((𝐴o 𝐵) ·o 𝐴))
48 eqid 2821 . . . . . 6 (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
49 ovex 7183 . . . . . 6 ((𝐴o 𝐵) ·o 𝐴) ∈ V
5047, 48, 49fvmpt 6763 . . . . 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 8134 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
5322, 52sylanl2 679 . . . . 5 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
5453fveq2d 6669 . . . 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 2870 . . 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 2866 . 2 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))
5740, 56oe0lem 8132 1 ((𝐴 ∈ On ∧ 𝐵𝑋) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3495  ∅c0 4291   ↦ cmpt 5139  Ord word 6185  Oncon0 6186  Lim wlim 6187  suc csuc 6188  ‘cfv 6350  (class class class)co 7150  reccrdg 8039  1oc1o 8089   ·o comu 8094   ↑o coe 8095 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-omul 8101  df-oexp 8102 This theorem is referenced by:  oesuc  8146  onesuc  8149
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