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Theorem oesuclem 8563
Description: Lemma for oesuc 8565. (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 7438 . . . 4 (𝐴 = ∅ → (𝐴o suc 𝐵) = (∅ ↑o suc 𝐵))
2 oesuclem.1 . . . . . . . 8 Lim 𝑋
3 limord 6444 . . . . . . . 8 (Lim 𝑋 → Ord 𝑋)
42, 3ax-mp 5 . . . . . . 7 Ord 𝑋
5 ordelord 6406 . . . . . . 7 ((Ord 𝑋𝐵𝑋) → Ord 𝐵)
64, 5mpan 690 . . . . . 6 (𝐵𝑋 → Ord 𝐵)
7 0elsuc 7855 . . . . . 6 (Ord 𝐵 → ∅ ∈ suc 𝐵)
86, 7syl 17 . . . . 5 (𝐵𝑋 → ∅ ∈ suc 𝐵)
9 limsuc 7870 . . . . . . 7 (Lim 𝑋 → (𝐵𝑋 ↔ suc 𝐵𝑋))
102, 9ax-mp 5 . . . . . 6 (𝐵𝑋 ↔ suc 𝐵𝑋)
11 ordelon 6408 . . . . . . . 8 ((Ord 𝑋 ∧ suc 𝐵𝑋) → suc 𝐵 ∈ On)
124, 11mpan 690 . . . . . . 7 (suc 𝐵𝑋 → suc 𝐵 ∈ On)
13 oe0m1 8559 . . . . . . 7 (suc 𝐵 ∈ On → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
1412, 13syl 17 . . . . . 6 (suc 𝐵𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
1510, 14sylbi 217 . . . . 5 (𝐵𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
168, 15mpbid 232 . . . 4 (𝐵𝑋 → (∅ ↑o suc 𝐵) = ∅)
171, 16sylan9eqr 2799 . . 3 ((𝐵𝑋𝐴 = ∅) → (𝐴o suc 𝐵) = ∅)
18 oveq1 7438 . . . . 5 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
19 id 22 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
2018, 19oveq12d 7449 . . . 4 (𝐴 = ∅ → ((𝐴o 𝐵) ·o 𝐴) = ((∅ ↑o 𝐵) ·o ∅))
21 ordelon 6408 . . . . . . 7 ((Ord 𝑋𝐵𝑋) → 𝐵 ∈ On)
224, 21mpan 690 . . . . . 6 (𝐵𝑋𝐵 ∈ On)
23 oveq2 7439 . . . . . . . . 9 (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
24 oe0m0 8558 . . . . . . . . . 10 (∅ ↑o ∅) = 1o
25 1on 8518 . . . . . . . . . 10 1o ∈ On
2624, 25eqeltri 2837 . . . . . . . . 9 (∅ ↑o ∅) ∈ On
2723, 26eqeltrdi 2849 . . . . . . . 8 (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
2827adantl 481 . . . . . . 7 ((𝐵𝑋𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
29 oe0m1 8559 . . . . . . . . . . 11 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
3022, 29syl 17 . . . . . . . . . 10 (𝐵𝑋 → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
3130biimpa 476 . . . . . . . . 9 ((𝐵𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
32 0elon 6438 . . . . . . . . 9 ∅ ∈ On
3331, 32eqeltrdi 2849 . . . . . . . 8 ((𝐵𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
3433adantll 714 . . . . . . 7 (((𝐵 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
3528, 34oe0lem 8551 . . . . . 6 ((𝐵 ∈ On ∧ 𝐵𝑋) → (∅ ↑o 𝐵) ∈ On)
3622, 35mpancom 688 . . . . 5 (𝐵𝑋 → (∅ ↑o 𝐵) ∈ On)
37 om0 8555 . . . . 5 ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ·o ∅) = ∅)
3836, 37syl 17 . . . 4 (𝐵𝑋 → ((∅ ↑o 𝐵) ·o ∅) = ∅)
3920, 38sylan9eqr 2799 . . 3 ((𝐵𝑋𝐴 = ∅) → ((𝐴o 𝐵) ·o 𝐴) = ∅)
4017, 39eqtr4d 2780 . 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 727 . . 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 217 . . . 4 (𝐵𝑋 → suc 𝐵 ∈ On)
44 oevn0 8553 . . . 4 (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
4543, 44sylanl2 681 . . 3 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
46 ovex 7464 . . . . 5 (𝐴o 𝐵) ∈ V
47 oveq1 7438 . . . . . 6 (𝑥 = (𝐴o 𝐵) → (𝑥 ·o 𝐴) = ((𝐴o 𝐵) ·o 𝐴))
48 eqid 2737 . . . . . 6 (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
49 ovex 7464 . . . . . 6 ((𝐴o 𝐵) ·o 𝐴) ∈ V
5047, 48, 49fvmpt 7016 . . . . 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 8553 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
5322, 52sylanl2 681 . . . . 5 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
5453fveq2d 6910 . . . 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 2791 . . 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 2787 . 2 (((𝐴 ∈ On ∧ 𝐵𝑋) ∧ ∅ ∈ 𝐴) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))
5740, 56oe0lem 8551 1 ((𝐴 ∈ On ∧ 𝐵𝑋) → (𝐴o suc 𝐵) = ((𝐴o 𝐵) ·o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cmpt 5225  Ord word 6383  Oncon0 6384  Lim wlim 6385  suc csuc 6386  cfv 6561  (class class class)co 7431  reccrdg 8449  1oc1o 8499   ·o comu 8504  o coe 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-omul 8511  df-oexp 8512
This theorem is referenced by:  oesuc  8565  onesuc  8568
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