Theorem omword2 8489

Description: An ordinal is less than or equal to its product with another. Lemma 3.12 of [Schloeder] p. 9. (Contributed by NM, 21-Dec-2004.)

Assertion

Ref	Expression
omword2	⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·o 𝐴))

Proof of Theorem omword2

Step	Hyp	Ref	Expression
1		om1r 8458	. . 3 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
2	1	ad2antrr 726	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o ·o 𝐴) = 𝐴)
3		eloni 6316	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordgt0ge1 8408	. . . . . 6 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵))
5	4	biimpa 476	. . . . 5 ⊢ ((Ord 𝐵 ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
6	3, 5	sylan 580	. . . 4 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
7	6	adantll 714	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
8		1on 8397	. . . . . 6 ⊢ 1o ∈ On
9		omwordri 8487	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
10	8, 9	mp3an1 1450	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
11	10	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
12	11	adantr 480	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
13	7, 12	mpd 15	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴))
14	2, 13	eqsstrrd 3965	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ∅c0 4280  Ord word 6305  Oncon0 6306  (class class class)co 7346  1_oc1o 8378   ·_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-rep 5215  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-1o 8385  df-oadd 8389  df-omul 8390

This theorem is referenced by:  omeulem1  8497  omabslem  8565  omabs  8566  omge2  43390