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Theorem nnmwordri 7580
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 7579 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
2 nnmcom 7570 . . . 4 ((𝐴 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 𝐶) = (𝐶 ·𝑜 𝐴))
323adant2 1072 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·𝑜 𝐶) = (𝐶 ·𝑜 𝐴))
4 nnmcom 7570 . . . 4 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 ·𝑜 𝐶) = (𝐶 ·𝑜 𝐵))
543adant1 1071 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐵 ·𝑜 𝐶) = (𝐶 ·𝑜 𝐵))
63, 5sseq12d 3596 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·𝑜 𝐶) ⊆ (𝐵 ·𝑜 𝐶) ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
71, 6sylibrd 247 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 1030   = wceq 1474  wcel 1976  wss 3539  (class class class)co 6527  ωcom 6934   ·𝑜 comu 7422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-oadd 7428  df-omul 7429
This theorem is referenced by:  omopthlem1  7599
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