Theorem omword2 8599

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 8568	. . 3 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
2	1	ad2antrr 724	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o ·o 𝐴) = 𝐴)
3		eloni 6382	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordgt0ge1 8518	. . . . . 6 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵))
5	4	biimpa 475	. . . . 5 ⊢ ((Ord 𝐵 ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
6	3, 5	sylan 578	. . . 4 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
7	6	adantll 712	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
8		1on 8503	. . . . . 6 ⊢ 1o ∈ On
9		omwordri 8597	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
10	8, 9	mp3an1 1444	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
11	10	ancoms 457	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
12	11	adantr 479	. . 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 4019	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ⊆ wss 3947  ∅c0 4324  Ord word 6371  Oncon0 6372  (class class class)co 7424  1_oc1o 8484   ·_o comu 8489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-oadd 8495  df-omul 8496

This theorem is referenced by:  omeulem1  8607  omabslem  8675  omabs  8676  omge2  42730