Theorem omword2 8630

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 8599	. . 3 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
2	1	ad2antrr 725	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o ·o 𝐴) = 𝐴)
3		eloni 6405	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordgt0ge1 8549	. . . . . 6 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵))
5	4	biimpa 476	. . . . 5 ⊢ ((Ord 𝐵 ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
6	3, 5	sylan 579	. . . 4 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
7	6	adantll 713	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
8		1on 8534	. . . . . 6 ⊢ 1o ∈ On
9		omwordri 8628	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
10	8, 9	mp3an1 1448	. . . . 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 4048	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  ∅c0 4352  Ord word 6394  Oncon0 6395  (class class class)co 7448  1_oc1o 8515   ·_o comu 8520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-omul 8527

This theorem is referenced by:  omeulem1  8638  omabslem  8706  omabs  8707  omge2  43260