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Theorem omcl 7564
Description: Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
omcl ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)

Proof of Theorem omcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6615 . . . 4 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
21eleq1d 2683 . . 3 (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 ∅) ∈ On))
3 oveq2 6615 . . . 4 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
43eleq1d 2683 . . 3 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 𝑦) ∈ On))
5 oveq2 6615 . . . 4 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
65eleq1d 2683 . . 3 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 suc 𝑦) ∈ On))
7 oveq2 6615 . . . 4 (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
87eleq1d 2683 . . 3 (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) ∈ On ↔ (𝐴 ·𝑜 𝐵) ∈ On))
9 om0 7545 . . . 4 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
10 0elon 5739 . . . 4 ∅ ∈ On
119, 10syl6eqel 2706 . . 3 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) ∈ On)
12 oacl 7563 . . . . . . 7 (((𝐴 ·𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On)
1312expcom 451 . . . . . 6 (𝐴 ∈ On → ((𝐴 ·𝑜 𝑦) ∈ On → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On))
1413adantr 481 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ∈ On → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On))
15 omsuc 7554 . . . . . 6 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
1615eleq1d 2683 . . . . 5 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 suc 𝑦) ∈ On ↔ ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ∈ On))
1714, 16sylibrd 249 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 suc 𝑦) ∈ On))
1817expcom 451 . . 3 (𝑦 ∈ On → (𝐴 ∈ On → ((𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 suc 𝑦) ∈ On)))
19 vex 3189 . . . . . 6 𝑥 ∈ V
20 iunon 7384 . . . . . 6 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ∈ On) → 𝑦𝑥 (𝐴 ·𝑜 𝑦) ∈ On)
2119, 20mpan 705 . . . . 5 (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ∈ On → 𝑦𝑥 (𝐴 ·𝑜 𝑦) ∈ On)
22 omlim 7561 . . . . . . 7 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = 𝑦𝑥 (𝐴 ·𝑜 𝑦))
2319, 22mpanr1 718 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝑥) → (𝐴 ·𝑜 𝑥) = 𝑦𝑥 (𝐴 ·𝑜 𝑦))
2423eleq1d 2683 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝑥) → ((𝐴 ·𝑜 𝑥) ∈ On ↔ 𝑦𝑥 (𝐴 ·𝑜 𝑦) ∈ On))
2521, 24syl5ibr 236 . . . 4 ((𝐴 ∈ On ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 𝑥) ∈ On))
2625expcom 451 . . 3 (Lim 𝑥 → (𝐴 ∈ On → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ∈ On → (𝐴 ·𝑜 𝑥) ∈ On)))
272, 4, 6, 8, 11, 18, 26tfinds3 7014 . 2 (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 ·𝑜 𝐵) ∈ On))
2827impcom 446 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  c0 3893   ciun 4487  Oncon0 5684  Lim wlim 5685  suc csuc 5686  (class class class)co 6607   +𝑜 coa 7505   ·𝑜 comu 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-oadd 7512  df-omul 7513
This theorem is referenced by:  oecl  7565  omordi  7594  omord2  7595  omcan  7597  omword  7598  omwordri  7600  om00  7603  om00el  7604  omlimcl  7606  odi  7607  omass  7608  oneo  7609  omeulem1  7610  omeulem2  7611  omopth2  7612  oeoelem  7626  oeoe  7627  oeeui  7630  oaabs2  7673  omxpenlem  8008  omxpen  8009  cantnfle  8515  cantnflt  8516  cantnflem1d  8532  cantnflem1  8533  cantnflem3  8535  cantnflem4  8536  cnfcomlem  8543  xpnum  8724  infxpenc  8788  dfac12lem2  8913
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