Theorem omcan 7649

Description: Left cancellation law for ordinal multiplication. Proposition 8.20 of [TakeutiZaring] p. 63 and its converse. (Contributed by NM, 14-Dec-2004.)

Assertion

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

Proof of Theorem omcan

Step	Hyp	Ref	Expression
1		omordi 7646	. . . . . . . . 9 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 ∈ 𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶)))
2	1	ex 450	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐵 ∈ 𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
3	2	ancoms 469	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵 ∈ 𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
4	3	3adant2 1080	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵 ∈ 𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶))))
5	4	imp 445	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 ∈ 𝐶 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶)))
6		omordi 7646	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶 ∈ 𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)))
7	6	ex 450	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐶 ∈ 𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
8	7	ancoms 469	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (𝐶 ∈ 𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
9	8	3adant3 1081	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐶 ∈ 𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
10	9	imp 445	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶 ∈ 𝐵 → (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)))
11	5, 10	orim12d 883	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
12	11	con3d 148	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵)) → ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
13		omcl 7616	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
14		eloni 5733	. . . . . . 7 ⊢ ((𝐴 ·𝑜 𝐵) ∈ On → Ord (𝐴 ·𝑜 𝐵))
15	13, 14	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ·𝑜 𝐵))
16		omcl 7616	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·𝑜 𝐶) ∈ On)
17		eloni 5733	. . . . . . 7 ⊢ ((𝐴 ·𝑜 𝐶) ∈ On → Ord (𝐴 ·𝑜 𝐶))
18	16, 17	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → Ord (𝐴 ·𝑜 𝐶))
19		ordtri3 5759	. . . . . 6 ⊢ ((Ord (𝐴 ·𝑜 𝐵) ∧ Ord (𝐴 ·𝑜 𝐶)) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ ¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
20	15, 18, 19	syl2an 494	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ∈ On ∧ 𝐶 ∈ On)) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ ¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
21	20	3impdi 1381	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ ¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
22	21	adantr 481	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ ¬ ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 𝐶) ∨ (𝐴 ·𝑜 𝐶) ∈ (𝐴 ·𝑜 𝐵))))
23		eloni 5733	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		eloni 5733	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
25		ordtri3 5759	. . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
26	23, 24, 25	syl2an 494	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
27	26	3adant1 1079	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
28	27	adantr 481	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
29	12, 22, 28	3imtr4d 283	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) → 𝐵 = 𝐶))
30		oveq2 6658	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶))
31	29, 30	impbid1 215	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∅c0 3915  Ord word 5722  Oncon0 5723  (class class class)co 6650   ·_𝑜 comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-omul 7565

This theorem is referenced by:  omword  7650  fin1a2lem4  9225