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Theorem omwordri 7885
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 6878 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
2 oveq2 6878 . . . . . 6 (𝑥 = ∅ → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 ∅))
31, 2sseq12d 3831 . . . . 5 (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅)))
4 oveq2 6878 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
5 oveq2 6878 . . . . . 6 (𝑥 = 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝑦))
64, 5sseq12d 3831 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦)))
7 oveq2 6878 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
8 oveq2 6878 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 suc 𝑦))
97, 8sseq12d 3831 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦)))
10 oveq2 6878 . . . . . 6 (𝑥 = 𝐶 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐶))
11 oveq2 6878 . . . . . 6 (𝑥 = 𝐶 → (𝐵 ·𝑜 𝑥) = (𝐵 ·𝑜 𝐶))
1210, 11sseq12d 3831 . . . . 5 (𝑥 = 𝐶 → ((𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥) ↔ (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
13 om0 7830 . . . . . . 7 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) = ∅)
14 0ss 4170 . . . . . . 7 ∅ ⊆ (𝐵 ·𝑜 ∅)
1513, 14syl6eqss 3852 . . . . . 6 (𝐴 ∈ On → (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅))
1615ad2antrr 708 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 ·𝑜 ∅) ⊆ (𝐵 ·𝑜 ∅))
17 omcl 7849 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
18173adant2 1154 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑦) ∈ On)
19 omcl 7849 . . . . . . . . . . . . . 14 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
20193adant1 1153 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 𝑦) ∈ On)
21 simp1 1159 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → 𝐴 ∈ On)
22 oawordri 7863 . . . . . . . . . . . . 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 1483 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴)))
2423imp 395 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦)) → ((𝐴 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴))
2524adantrl 698 . . . . . . . . . 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 7862 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐵 ·𝑜 𝑦) ∈ On) → (𝐴𝐵 ↔ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
2720, 26syld3an3 1521 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴𝐵 ↔ ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵)))
2827biimpa 464 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ 𝐴𝐵) → ((𝐵 ·𝑜 𝑦) +𝑜 𝐴) ⊆ ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
2928adantrr 699 . . . . . . . . . 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 3808 . . . . . . . . 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 7839 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
32313adant2 1154 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
3332adantr 468 . . . . . . . . 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 7839 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
35343adant1 1153 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 ·𝑜 suc 𝑦) = ((𝐵 ·𝑜 𝑦) +𝑜 𝐵))
3635adantr 468 . . . . . . . . 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 3845 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑦 ∈ On) ∧ (𝐴𝐵 ∧ (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦))) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))
3837exp520 1459 . . . . . . 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 412 . . . . 5 (𝑦 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ((𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 suc 𝑦) ⊆ (𝐵 ·𝑜 suc 𝑦))))
41 vex 3394 . . . . . . . 8 𝑥 ∈ V
42 ss2iun 4728 . . . . . . . . . 10 (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → 𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ 𝑦𝑥 (𝐵 ·𝑜 𝑦))
43 omlim 7846 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐴 ·𝑜 𝑥) = 𝑦𝑥 (𝐴 ·𝑜 𝑦))
4443ad2ant2rl 746 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐴 ·𝑜 𝑥) = 𝑦𝑥 (𝐴 ·𝑜 𝑦))
45 omlim 7846 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (𝐵 ·𝑜 𝑥) = 𝑦𝑥 (𝐵 ·𝑜 𝑦))
4645adantl 469 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ (𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥))) → (𝐵 ·𝑜 𝑥) = 𝑦𝑥 (𝐵 ·𝑜 𝑦))
4744, 46sseq12d 3831 . . . . . . . . . 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 237 . . . . . . . . 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 661 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥)))
5041, 49mpanr1 686 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥)))
5150expcom 400 . . . . . 6 (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦𝑥 (𝐴 ·𝑜 𝑦) ⊆ (𝐵 ·𝑜 𝑦) → (𝐴 ·𝑜 𝑥) ⊆ (𝐵 ·𝑜 𝑥))))
5251adantrd 481 . . . . 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 7290 . . . 4 (𝐶 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
5453expd 402 . . 3 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶))))
55543impib 1137 . 2 ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
56553coml 1150 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wral 3096  Vcvv 3391  wss 3769  c0 4116   ciun 4712  Oncon0 5936  Lim wlim 5937  suc csuc 5938  (class class class)co 6870   +𝑜 coa 7789   ·𝑜 comu 7790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-oadd 7796  df-omul 7797
This theorem is referenced by:  omword2  7887  oewordri  7905  oeordsuc  7907
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