Theorem omord 7535

Description: Ordering property of ordinal multiplication. Proposition 8.19 of [TakeutiZaring] p. 63. (Contributed by NM, 14-Dec-2004.)

Assertion

Ref	Expression
omord	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Proof of Theorem omord

Step	Hyp	Ref	Expression
1		omord2 7534	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
2	1	ex 449	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴 ∈ 𝐵 ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
3	2	pm5.32rd 670	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∧ ∅ ∈ 𝐶)))
4		simpl 472	. . 3 ⊢ (((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))
5		ne0i 3880	. . . . . . . 8 ⊢ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (𝐶 ·𝑜 𝐵) ≠ ∅)
6		om0r 7506	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
7		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
8	7	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝐶 = ∅ → ((𝐶 ·𝑜 𝐵) = ∅ ↔ (∅ ·𝑜 𝐵) = ∅))
9	6, 8	syl5ibrcom 236	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = ∅))
10	9	necon3d 2803	. . . . . . . 8 ⊢ (𝐵 ∈ On → ((𝐶 ·𝑜 𝐵) ≠ ∅ → 𝐶 ≠ ∅))
11	5, 10	syl5 33	. . . . . . 7 ⊢ (𝐵 ∈ On → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
12	11	adantr 480	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
13		on0eln0 5697	. . . . . . 7 ⊢ (𝐶 ∈ On → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
14	13	adantl 481	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
15	12, 14	sylibrd 248	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
16	15	3adant1 1072	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
17	16	ancld 574	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∧ ∅ ∈ 𝐶)))
18	4, 17	impbid2 215	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
19	3, 18	bitrd 267	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  Oncon0 5640  (class class class)co 6549   ·_𝑜 comu 7445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452

This theorem is referenced by:  omlimcl  7545  oneo  7548