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Theorem oesuclem 8546
Description: Lemma for oesuc 8548. (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 7426 . . . 4 (𝐴 = ∅ → (𝐴o suc 𝐵) = (∅ ↑o suc 𝐵))
2 oesuclem.1 . . . . . . . 8 Lim 𝑋
3 limord 6431 . . . . . . . 8 (Lim 𝑋 → Ord 𝑋)
42, 3ax-mp 5 . . . . . . 7 Ord 𝑋
5 ordelord 6393 . . . . . . 7 ((Ord 𝑋𝐵𝑋) → Ord 𝐵)
64, 5mpan 688 . . . . . 6 (𝐵𝑋 → Ord 𝐵)
7 0elsuc 7839 . . . . . 6 (Ord 𝐵 → ∅ ∈ suc 𝐵)
86, 7syl 17 . . . . 5 (𝐵𝑋 → ∅ ∈ suc 𝐵)
9 limsuc 7854 . . . . . . 7 (Lim 𝑋 → (𝐵𝑋 ↔ suc 𝐵𝑋))
102, 9ax-mp 5 . . . . . 6 (𝐵𝑋 ↔ suc 𝐵𝑋)
11 ordelon 6395 . . . . . . . 8 ((Ord 𝑋 ∧ suc 𝐵𝑋) → suc 𝐵 ∈ On)
124, 11mpan 688 . . . . . . 7 (suc 𝐵𝑋 → suc 𝐵 ∈ On)
13 oe0m1 8542 . . . . . . 7 (suc 𝐵 ∈ On → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
1412, 13syl 17 . . . . . 6 (suc 𝐵𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
1510, 14sylbi 216 . . . . 5 (𝐵𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
168, 15mpbid 231 . . . 4 (𝐵𝑋 → (∅ ↑o suc 𝐵) = ∅)
171, 16sylan9eqr 2787 . . 3 ((𝐵𝑋𝐴 = ∅) → (𝐴o suc 𝐵) = ∅)
18 oveq1 7426 . . . . 5 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
19 id 22 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
2018, 19oveq12d 7437 . . . 4 (𝐴 = ∅ → ((𝐴o 𝐵) ·o 𝐴) = ((∅ ↑o 𝐵) ·o ∅))
21 ordelon 6395 . . . . . . 7 ((Ord 𝑋𝐵𝑋) → 𝐵 ∈ On)
224, 21mpan 688 . . . . . 6 (𝐵𝑋𝐵 ∈ On)
23 oveq2 7427 . . . . . . . . 9 (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
24 oe0m0 8541 . . . . . . . . . 10 (∅ ↑o ∅) = 1o
25 1on 8499 . . . . . . . . . 10 1o ∈ On
2624, 25eqeltri 2821 . . . . . . . . 9 (∅ ↑o ∅) ∈ On
2723, 26eqeltrdi 2833 . . . . . . . 8 (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
2827adantl 480 . . . . . . 7 ((𝐵𝑋𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
29 oe0m1 8542 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
3022, 29syl 17 . . . . . . . . . 10 (𝐵𝑋 → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
3130biimpa 475 . . . . . . . . 9 ((𝐵𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
32 0elon 6425 . . . . . . . . 9 ∅ ∈ On
3331, 32eqeltrdi 2833 . . . . . . . 8 ((𝐵𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
3433adantll 712 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
3528, 34oe0lem 8534 . . . . . 6 ((𝐵 ∈ On ∧ 𝐵𝑋) → (∅ ↑o 𝐵) ∈ On)
3622, 35mpancom 686 . . . . 5 (𝐵𝑋 → (∅ ↑o 𝐵) ∈ On)
37 om0 8538 . . . . 5 ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ·o ∅) = ∅)
3836, 37syl 17 . . . 4 (𝐵𝑋 → ((∅ ↑o 𝐵) ·o ∅) = ∅)
3920, 38sylan9eqr 2787 . . 3 ((𝐵𝑋𝐴 = ∅) → ((𝐴o 𝐵) ·o 𝐴) = ∅)
4017, 39eqtr4d 2768 . 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 216 . . . 4 (𝐵𝑋 → suc 𝐵 ∈ On)
44 oevn0 8536 . . . 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 7452 . . . . 5 (𝐴o 𝐵) ∈ V
47 oveq1 7426 . . . . . 6 (𝑥 = (𝐴o 𝐵) → (𝑥 ·o 𝐴) = ((𝐴o 𝐵) ·o 𝐴))
48 eqid 2725 . . . . . 6 (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
49 ovex 7452 . . . . . 6 ((𝐴o 𝐵) ·o 𝐴) ∈ V
5047, 48, 49fvmpt 7004 . . . . 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 8536 . . . . . 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 6900 . . . 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, 54eqtr3id 2779 . . 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 2775 . 2 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))
5740, 56oe0lem 8534 1 ((𝐴 ∈ On ∧ 𝐵𝑋) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  c0 4322  cmpt 5232  Ord word 6370  Oncon0 6371  Lim wlim 6372  suc csuc 6373  cfv 6549  (class class class)co 7419  reccrdg 8430  1oc1o 8480   ·o comu 8485  o coe 8486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-omul 8492  df-oexp 8493
This theorem is referenced by:  oesuc  8548  onesuc  8551
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