Theorem omcan 8533

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 8530	. . . . . . . . 9 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶)))
2	1	ex 412	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
3	2	ancoms 458	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
4	3	3adant2 1131	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶))))
5	4	imp 406	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 ∈ 𝐶 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o 𝐶)))
6		omordi 8530	. . . . . . . . 9 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶 ∈ 𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵)))
7	6	ex 412	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ∈ 𝐴 → (𝐶 ∈ 𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
8	7	ancoms 458	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐴 → (𝐶 ∈ 𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
9	8	3adant3 1132	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐴 → (𝐶 ∈ 𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵))))
10	9	imp 406	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐶 ∈ 𝐵 → (𝐴 ·o 𝐶) ∈ (𝐴 ·o 𝐵)))
11	5, 10	orim12d 966	. . . 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 152	. . 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 8500	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
14		eloni 6342	. . . . . . 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 8500	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o 𝐶) ∈ On)
17		eloni 6342	. . . . . . 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 6368	. . . . . 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 596	. . . . 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 480	. . 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 6342	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		eloni 6342	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
25		ordtri3 6368	. . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
26	23, 24, 25	syl2an 596	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
27	26	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
28	27	adantr 480	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
29	12, 22, 28	3imtr4d 294	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) → 𝐵 = 𝐶))
30		oveq2 7395	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐶))
31	29, 30	impbid1 225	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∅c0 4296  Ord word 6331  Oncon0 6332  (class class class)co 7387   ·_o comu 8432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-oadd 8438  df-omul 8439

This theorem is referenced by:  omword  8534  fin1a2lem4  10356