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Theorem nnmwordi 7660
Description: Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
nnmwordi ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Proof of Theorem nnmwordi
StepHypRef Expression
1 nnmword 7658 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
21biimpd 219 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
32ex 450 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))))
4 nnord 7020 . . . . . 6 (𝐶 ∈ ω → Ord 𝐶)
5 ord0eln0 5738 . . . . . . 7 (Ord 𝐶 → (∅ ∈ 𝐶𝐶 ≠ ∅))
65necon2bbid 2833 . . . . . 6 (Ord 𝐶 → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
74, 6syl 17 . . . . 5 (𝐶 ∈ ω → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
873ad2ant3 1082 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
9 ssid 3603 . . . . . . 7 ∅ ⊆ ∅
10 nnm0r 7635 . . . . . . . . 9 (𝐴 ∈ ω → (∅ ·𝑜 𝐴) = ∅)
1110adantr 481 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∅ ·𝑜 𝐴) = ∅)
12 nnm0r 7635 . . . . . . . . 9 (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = ∅)
1312adantl 482 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∅ ·𝑜 𝐵) = ∅)
1411, 13sseq12d 3613 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵) ↔ ∅ ⊆ ∅))
159, 14mpbiri 248 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵))
16 oveq1 6611 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) = (∅ ·𝑜 𝐴))
17 oveq1 6611 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
1816, 17sseq12d 3613 . . . . . 6 (𝐶 = ∅ → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ (∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵)))
1915, 18syl5ibrcom 237 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
20193adant3 1079 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
218, 20sylbird 250 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (¬ ∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
2221a1dd 50 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (¬ ∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))))
233, 22pm2.61d 170 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wss 3555  c0 3891  Ord word 5681  (class class class)co 6604  ωcom 7012   ·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-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-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-omul 7510
This theorem is referenced by:  nnmwordri  7661  omopthlem1  7680
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