Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  omlim Structured version   Visualization version   GIF version

Theorem omlim 7500
 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 5705 . . 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 7416 . . . 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 7479 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵))
7 onelon 5665 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 omv 7479 . . . . . . . 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 678 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥𝐵) → (𝐴 ·𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
1110iuneq2dv 4478 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝑥𝐵 (𝐴 ·𝑜 𝑥) = 𝑥𝐵 (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
126, 11eqeq12d 2625 . . . 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 749 . . 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 246 . 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ∪ ciun 4455   ↦ cmpt 4643  Oncon0 5640  Lim wlim 5641  ‘cfv 5804  (class class class)co 6549  reccrdg 7392   +𝑜 coa 7444   ·𝑜 comu 7445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-omul 7452 This theorem is referenced by:  omcl  7503  om0r  7506  om1r  7510  omordi  7533  omwordri  7539  omordlim  7544  omlimcl  7545  odi  7546  omass  7547  omeulem1  7549  oeoalem  7563  oeoelem  7565  omabslem  7613  omabs  7614
 Copyright terms: Public domain W3C validator