Theorem omcan 8226

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 8223	. . . . . . . . 9 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶)))
2	1	ex 416	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
3	2	ancoms 462	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
4	3	3adant2 1132	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
5	4	imp 410	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶)))
6		omordi 8223	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶 ∈ 𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵)))
7	6	ex 416	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐶 ∈ 𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
8	7	ancoms 462	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (𝐶 ∈ 𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
9	8	3adant3 1133	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐶 ∈ 𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
10	9	imp 410	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶 ∈ 𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵)))
11	5, 10	orim12d 964	. . . 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 155	. . 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 8192	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
14		eloni 6182	. . . . . . 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 8192	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o 𝐶) ∈ On)
17		eloni 6182	. . . . . . 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 6208	. . . . . 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 599	. . . . 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 1351	. . . 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 484	. . 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 6182	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		eloni 6182	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
25		ordtri3 6208	. . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
26	23, 24, 25	syl2an 599	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
27	26	3adant1 1131	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
28	27	adantr 484	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
29	12, 22, 28	3imtr4d 297	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
30		oveq2 7178	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶))
31	29, 30	impbid1 228	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∅c0 4211  Ord word 6171  Oncon0 6172  (class class class)co 7170   ·_o comu 8129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-oadd 8135  df-omul 8136

This theorem is referenced by:  omword  8227  fin1a2lem4  9903