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Theorem nnmwordi 8236
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 8234 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
21biimpd 232 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
32ex 416 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))))
4 nnord 7563 . . . . . 6 (𝐶 ∈ ω → Ord 𝐶)
5 ord0eln0 6218 . . . . . . 7 (Ord 𝐶 → (∅ ∈ 𝐶𝐶 ≠ ∅))
65necon2bbid 3050 . . . . . 6 (Ord 𝐶 → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
74, 6syl 17 . . . . 5 (𝐶 ∈ ω → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
873ad2ant3 1132 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
9 ssid 3965 . . . . . . 7 ∅ ⊆ ∅
10 nnm0r 8211 . . . . . . . . 9 (𝐴 ∈ ω → (∅ ·o 𝐴) = ∅)
1110adantr 484 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∅ ·o 𝐴) = ∅)
12 nnm0r 8211 . . . . . . . . 9 (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
1312adantl 485 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∅ ·o 𝐵) = ∅)
1411, 13sseq12d 3976 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ·o 𝐴) ⊆ (∅ ·o 𝐵) ↔ ∅ ⊆ ∅))
159, 14mpbiri 261 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵))
16 oveq1 7137 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·o 𝐴) = (∅ ·o 𝐴))
17 oveq1 7137 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·o 𝐵) = (∅ ·o 𝐵))
1816, 17sseq12d 3976 . . . . . 6 (𝐶 = ∅ → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵)))
1915, 18syl5ibrcom 250 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 = ∅ → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
20193adant3 1129 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 = ∅ → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
218, 20sylbird 263 . . 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 182 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wss 3910  c0 4266  Ord word 6163  (class class class)co 7130  ωcom 7555   ·o comu 8075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-oadd 8081  df-omul 8082
This theorem is referenced by:  nnmwordri  8237  omopthlem1  8257
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