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Theorem nnmwordri 7603
 Description: Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
nnmwordri ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))

Proof of Theorem nnmwordri
StepHypRef Expression
1 nnmwordi 7602 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
2 nnmcom 7593 . . . 4 ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 𝐶) = (𝐶 ·𝑜 𝐴))
323adant2 1073 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 𝐶) = (𝐶 ·𝑜 𝐴))
4 nnmcom 7593 . . . 4 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 ·𝑜 𝐶) = (𝐶 ·𝑜 𝐵))
543adant1 1072 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 ·𝑜 𝐶) = (𝐶 ·𝑜 𝐵))
63, 5sseq12d 3597 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶) ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
71, 6sylibrd 248 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  (class class class)co 6549  ωcom 6957   ·𝑜 comu 7445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452 This theorem is referenced by:  omopthlem1  7622
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