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Theorem omlim 7658
Description: Ordinal multiplication with a limit ordinal. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 3-Aug-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
omlim ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem omlim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 limelon 5826 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 simpr 476 . . 3 ((𝐵𝐶 ∧ Lim 𝐵) → Lim 𝐵)
31, 2jca 553 . 2 ((𝐵𝐶 ∧ Lim 𝐵) → (𝐵 ∈ On ∧ Lim 𝐵))
4 rdglim2a 7574 . . . 4 ((𝐵 ∈ On ∧ Lim 𝐵) → (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
54adantl 481 . . 3 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
6 omv 7637 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵))
7 onelon 5786 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 omv 7637 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
97, 8sylan2 490 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥𝐵)) → (𝐴 ·𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
109anassrs 681 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥𝐵) → (𝐴 ·𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
1110iuneq2dv 4574 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝑥𝐵 (𝐴 ·𝑜 𝑥) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
126, 11eqeq12d 2666 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥) ↔ (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
1312adantrr 753 . . 3 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → ((𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥) ↔ (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
145, 13mpbird 247 . 2 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ Lim 𝐵)) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥))
153, 14sylan2 490 1 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948   ciun 4552  cmpt 4762  Oncon0 5761  Lim wlim 5762  cfv 5926  (class class class)co 6690  reccrdg 7550   +𝑜 coa 7602   ·𝑜 comu 7603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-omul 7610
This theorem is referenced by:  omcl  7661  om0r  7664  om1r  7668  omordi  7691  omwordri  7697  omordlim  7702  omlimcl  7703  odi  7704  omass  7705  omeulem1  7707  oeoalem  7721  oeoelem  7723  omabslem  7771  omabs  7772
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