Theorem nnmwordi 8550

Description: Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
nnmwordi	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))

Proof of Theorem nnmwordi

Step	Hyp	Ref	Expression
1		nnmword 8548	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
2	1	biimpd 229	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
3	2	ex 412	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))))
4		nnord 7804	. . . . . 6 ⊢ (𝐶 ∈ ω → Ord 𝐶)
5		ord0eln0 6362	. . . . . . 7 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
6	5	necon2bbid 2971	. . . . . 6 ⊢ (Ord 𝐶 → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
7	4, 6	syl 17	. . . . 5 ⊢ (𝐶 ∈ ω → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
8	7	3ad2ant3 1135	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
9		ssid 3952	. . . . . . 7 ⊢ ∅ ⊆ ∅
10		nnm0r 8525	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (∅ ·o 𝐴) = ∅)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∅ ·o 𝐴) = ∅)
12		nnm0r 8525	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
13	12	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∅ ·o 𝐵) = ∅)
14	11, 13	sseq12d 3963	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ·o 𝐴) ⊆ (∅ ·o 𝐵) ↔ ∅ ⊆ ∅))
15	9, 14	mpbiri 258	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵))
16		oveq1 7353	. . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ·o 𝐴) = (∅ ·o 𝐴))
17		oveq1 7353	. . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ·o 𝐵) = (∅ ·o 𝐵))
18	16, 17	sseq12d 3963	. . . . . 6 ⊢ (𝐶 = ∅ → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵)))
19	15, 18	syl5ibrcom 247	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 = ∅ → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
20	19	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐶 = ∅ → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
21	8, 20	sylbird 260	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (¬ ∅ ∈ 𝐶 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
22	21	a1dd 50	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (¬ ∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))))
23	3, 22	pm2.61d 179	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ∅c0 4280  Ord word 6305  (class class class)co 7346  ωcom 7796   ·_o comu 8383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-oadd 8389  df-omul 8390

This theorem is referenced by:  nnmwordri  8551  omopthlem1  8574