Theorem omcan 8495

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 8492	. . . . . . . . 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 1132	. . . . . 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 8492	. . . . . . . . 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 1133	. . . . . 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 967	. . . 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 8462	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
14		eloni 6325	. . . . . . 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 8462	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·o 𝐶) ∈ On)
17		eloni 6325	. . . . . . 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 6351	. . . . . 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 597	. . . . 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 1352	. . . 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 6325	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		eloni 6325	. . . . . 6 ⊢ (𝐶 ∈ On → Ord 𝐶)
25		ordtri3 6351	. . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵)))
26	23, 24, 25	syl2an 597	. . . . 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 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 7366	. 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 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∅c0 4274  Ord word 6314  Oncon0 6315  (class class class)co 7358   ·_o comu 8394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-oadd 8400  df-omul 8401

This theorem is referenced by:  omword  8496  fin1a2lem4  10314