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Theorem omwordri 7597
Description: Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Dec-2004.)
Assertion
Ref Expression
omwordri ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))

Proof of Theorem omwordri
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6612 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
2 oveq2 6612 . . . . . 6 (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅))
31, 2sseq12d 3613 . . . . 5 (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅)))
4 oveq2 6612 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
5 oveq2 6612 . . . . . 6 (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦))
64, 5sseq12d 3613 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦)))
7 oveq2 6612 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
8 oveq2 6612 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦))
97, 8sseq12d 3613 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦)))
10 oveq2 6612 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐶))
11 oveq2 6612 . . . . . 6 (𝑥 = 𝐶 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐶))
1210, 11sseq12d 3613 . . . . 5 (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
13 om0 7542 . . . . . . 7 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
14 0ss 3944 . . . . . . 7 ∅ ⊆ (𝐵 ·𝑜 ∅)
1513, 14syl6eqss 3634 . . . . . 6 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅))
1615ad2antrr 761 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅))
17 omcl 7561 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
18173adant2 1078 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
19 omcl 7561 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
20193adant1 1077 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
21 simp1 1059 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
22 oawordri 7575 . . . . . . . . . . . . 13 (((𝐴 ·𝑜 𝑦) ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴)))
2318, 20, 21, 22syl3anc 1323 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴)))
2423imp 445 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴))
2524adantrl 751 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴))
26 oaword 7574 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On) → (𝐴𝐵 ↔ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
2720, 26syld3an3 1368 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝐵 ↔ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
2827biimpa 501 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝐵) → ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
2928adantrr 752 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
3025, 29sstrd 3593 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
31 omsuc 7551 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
32313adant2 1078 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
3332adantr 481 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
34 omsuc 7551 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
35343adant1 1077 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
3635adantr 481 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
3730, 33, 363sstr4d 3627 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))
3837exp520 1285 . . . . . . 7 (𝐴 ∈ On → (𝐵 ∈ On → (𝑦 ∈ On → (𝐴𝐵 → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))))))
3938com3r 87 . . . . . 6 (𝑦 ∈ On → (𝐴 ∈ On → (𝐵 ∈ On → (𝐴𝐵 → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))))))
4039imp4c 616 . . . . 5 (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))))
41 vex 3189 . . . . . . . 8 𝑥 ∈ V
42 ss2iun 4502 . . . . . . . . . 10 (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → 𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ 𝑦𝑥 (𝐵 ·𝑜 𝑦))
43 omlim 7558 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = 𝑦𝑥 (𝐴 ·𝑜 𝑦))
4443ad2ant2rl 784 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐴 ·𝑜 𝑥) = 𝑦𝑥 (𝐴 ·𝑜 𝑦))
45 omlim 7558 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·𝑜 𝑥) = 𝑦𝑥 (𝐵 ·𝑜 𝑦))
4645adantl 482 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐵 ·𝑜 𝑥) = 𝑦𝑥 (𝐵 ·𝑜 𝑦))
4744, 46sseq12d 3613 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ 𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ 𝑦𝑥 (𝐵 ·𝑜 𝑦)))
4842, 47syl5ibr 236 . . . . . . . . 9 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥)))
4948anandirs 873 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥)))
5041, 49mpanr1 718 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥)))
5150expcom 451 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥))))
5251adantrd 484 . . . . 5 (Lim 𝑥 → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥))))
533, 6, 9, 12, 16, 40, 52tfinds3 7011 . . . 4 (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
5453expd 452 . . 3 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶))))
55543impib 1259 . 2 ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
56553coml 1269 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3555  c0 3891   ciun 4485  Oncon0 5682  Lim wlim 5683  suc csuc 5684  (class class class)co 6604   +𝑜 coa 7502   ·𝑜 comu 7503
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-omul 7510
This theorem is referenced by:  omword2  7599  oewordri  7617  oeordsuc  7619
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