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Theorem omwordri 7821
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 6821 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
2 oveq2 6821 . . . . . 6 (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅))
31, 2sseq12d 3775 . . . . 5 (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅)))
4 oveq2 6821 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
5 oveq2 6821 . . . . . 6 (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦))
64, 5sseq12d 3775 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦)))
7 oveq2 6821 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
8 oveq2 6821 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦))
97, 8sseq12d 3775 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦)))
10 oveq2 6821 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐶))
11 oveq2 6821 . . . . . 6 (𝑥 = 𝐶 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐶))
1210, 11sseq12d 3775 . . . . 5 (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
13 om0 7766 . . . . . . 7 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
14 0ss 4115 . . . . . . 7 ∅ ⊆ (𝐵 ·𝑜 ∅)
1513, 14syl6eqss 3796 . . . . . 6 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅))
1615ad2antrr 764 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅))
17 omcl 7785 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
18173adant2 1126 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
19 omcl 7785 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
20193adant1 1125 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
21 simp1 1131 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
22 oawordri 7799 . . . . . . . . . . . . 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 1477 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴)))
2423imp 444 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴))
2524adantrl 754 . . . . . . . . . 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 7798 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On) → (𝐴𝐵 ↔ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
2720, 26syld3an3 1516 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝐵 ↔ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
2827biimpa 502 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝐵) → ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
2928adantrr 755 . . . . . . . . . 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 3754 . . . . . . . . 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 7775 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
32313adant2 1126 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
3332adantr 472 . . . . . . . . 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 7775 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
35343adant1 1125 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
3635adantr 472 . . . . . . . . 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 3789 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))
3837exp520 1451 . . . . . . 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 618 . . . . 5 (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))))
41 vex 3343 . . . . . . . 8 𝑥 ∈ V
42 ss2iun 4688 . . . . . . . . . 10 (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → 𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ 𝑦𝑥 (𝐵 ·𝑜 𝑦))
43 omlim 7782 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = 𝑦𝑥 (𝐴 ·𝑜 𝑦))
4443ad2ant2rl 802 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐴 ·𝑜 𝑥) = 𝑦𝑥 (𝐴 ·𝑜 𝑦))
45 omlim 7782 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·𝑜 𝑥) = 𝑦𝑥 (𝐵 ·𝑜 𝑦))
4645adantl 473 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐵 ·𝑜 𝑥) = 𝑦𝑥 (𝐵 ·𝑜 𝑦))
4744, 46sseq12d 3775 . . . . . . . . . 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 909 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥)))
5041, 49mpanr1 721 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥)))
5150expcom 450 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥))))
5251adantrd 485 . . . . 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 7229 . . . 4 (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
5453expd 451 . . 3 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶))))
55543impib 1109 . 2 ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
56553coml 1122 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 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715  c0 4058   ciun 4672  Oncon0 5884  Lim wlim 5885  suc csuc 5886  (class class class)co 6813   +𝑜 coa 7726   ·𝑜 comu 7727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733  df-omul 7734
This theorem is referenced by:  omword2  7823  oewordri  7841  oeordsuc  7843
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