Theorem omwordi 7805

Description: Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)

Assertion

Ref	Expression
omwordi	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Proof of Theorem omwordi

Step	Hyp	Ref	Expression
1		omword 7804	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
2	1	biimpd 219	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
3	2	ex 397	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))))
4		eloni 5876	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
5		ord0eln0 5922	. . . . . . 7 ⊢ (Ord 𝐶 → (∅ ∈ 𝐶 ↔ 𝐶 ≠ ∅))
6	5	necon2bbid 2986	. . . . . 6 ⊢ (Ord 𝐶 → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
7	4, 6	syl 17	. . . . 5 ⊢ (𝐶 ∈ On → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
8	7	3ad2ant3 1129	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
9		ssid 3773	. . . . . . 7 ⊢ ∅ ⊆ ∅
10		om0r 7773	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅)
11	10	adantr 466	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐴) = ∅)
12		om0r 7773	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
13	12	adantl 467	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐵) = ∅)
14	11, 13	sseq12d 3783	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵) ↔ ∅ ⊆ ∅))
15	9, 14	mpbiri 248	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵))
16		oveq1 6800	. . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) = (∅ ·𝑜 𝐴))
17		oveq1 6800	. . . . . . 7 ⊢ (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
18	16, 17	sseq12d 3783	. . . . . 6 ⊢ (𝐶 = ∅ → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ (∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵)))
19	15, 18	syl5ibrcom 237	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
20	19	3adant3 1126	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
21	8, 20	sylbird 250	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
22	21	a1dd 50	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ∅ ∈ 𝐶 → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))))
23	3, 22	pm2.61d 171	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ⊆ wss 3723  ∅c0 4063  Ord word 5865  Oncon0 5866  (class class class)co 6793   ·_𝑜 comu 7711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-omul 7718

This theorem is referenced by:  omword1  7807  omass  7814  omeulem1  7816  oewordri  7826  oeoalem  7830  oeeui  7836  oaabs2  7879  omxpenlem  8217  cantnflt  8733  cantnflem1d  8749