Theorem omeulem2 8203

Description: Lemma for omeu 8205: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
omeulem2	⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))

Proof of Theorem omeulem2

Step	Hyp	Ref	Expression
1		simp3l 1197	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐷 ∈ On)
2		eloni 6196	. . . . . 6 ⊢ (𝐷 ∈ On → Ord 𝐷)
3		ordsucss 7527	. . . . . 6 ⊢ (Ord 𝐷 → (𝐵 ∈ 𝐷 → suc 𝐵 ⊆ 𝐷))
4	1, 2, 3	3syl 18	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐵 ∈ 𝐷 → suc 𝐵 ⊆ 𝐷))
5		simp2l 1195	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐵 ∈ On)
6		suceloni 7522	. . . . . . 7 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
7	5, 6	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → suc 𝐵 ∈ On)
8		simp1l 1193	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐴 ∈ On)
9		simp1r 1194	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐴 ≠ ∅)
10		on0eln0 6241	. . . . . . . 8 ⊢ (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
11	8, 10	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
12	9, 11	mpbird 259	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ∅ ∈ 𝐴)
13		omword 8190	. . . . . 6 ⊢ (((suc 𝐵 ∈ On ∧ 𝐷 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (suc 𝐵 ⊆ 𝐷 ↔ (𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷)))
14	7, 1, 8, 12, 13	syl31anc 1369	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (suc 𝐵 ⊆ 𝐷 ↔ (𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷)))
15	4, 14	sylibd 241	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐵 ∈ 𝐷 → (𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷)))
16		omcl 8155	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ·o 𝐷) ∈ On)
17	8, 1, 16	syl2anc 586	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐴 ·o 𝐷) ∈ On)
18		simp3r 1198	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐸 ∈ 𝐴)
19		onelon 6211	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐸 ∈ 𝐴) → 𝐸 ∈ On)
20	8, 18, 19	syl2anc 586	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐸 ∈ On)
21		oaword1 8172	. . . . . 6 ⊢ (((𝐴 ·o 𝐷) ∈ On ∧ 𝐸 ∈ On) → (𝐴 ·o 𝐷) ⊆ ((𝐴 ·o 𝐷) +o 𝐸))
22		sstr 3975	. . . . . . 7 ⊢ (((𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷) ∧ (𝐴 ·o 𝐷) ⊆ ((𝐴 ·o 𝐷) +o 𝐸)) → (𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸))
23	22	expcom 416	. . . . . 6 ⊢ ((𝐴 ·o 𝐷) ⊆ ((𝐴 ·o 𝐷) +o 𝐸) → ((𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷) → (𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸)))
24	21, 23	syl 17	. . . . 5 ⊢ (((𝐴 ·o 𝐷) ∈ On ∧ 𝐸 ∈ On) → ((𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷) → (𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸)))
25	17, 20, 24	syl2anc 586	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷) → (𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸)))
26	15, 25	syld 47	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐵 ∈ 𝐷 → (𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸)))
27		simp2r 1196	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
28		onelon 6211	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On)
29	8, 27, 28	syl2anc 586	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐶 ∈ On)
30		omcl 8155	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
31	8, 5, 30	syl2anc 586	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐴 ·o 𝐵) ∈ On)
32		oaord 8167	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (𝐶 ∈ 𝐴 ↔ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴)))
33	32	biimpa 479	. . . . 5 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) ∧ 𝐶 ∈ 𝐴) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴))
34	29, 8, 31, 27, 33	syl31anc 1369	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴))
35		omsuc 8145	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))
36	8, 5, 35	syl2anc 586	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))
37	34, 36	eleqtrrd 2916	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ (𝐴 ·o suc 𝐵))
38		ssel 3961	. . 3 ⊢ ((𝐴 ·o suc 𝐵) ⊆ ((𝐴 ·o 𝐷) +o 𝐸) → (((𝐴 ·o 𝐵) +o 𝐶) ∈ (𝐴 ·o suc 𝐵) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
39	26, 37, 38	syl6ci 71	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐵 ∈ 𝐷 → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
40		simpr 487	. . . . 5 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → 𝐶 ∈ 𝐸)
41		oaord 8167	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (𝐶 ∈ 𝐸 ↔ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐸)))
42	40, 41	syl5ib 246	. . . 4 ⊢ ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐸)))
43		oveq2 7158	. . . . . . 7 ⊢ (𝐵 = 𝐷 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐷))
44	43	oveq1d 7165	. . . . . 6 ⊢ (𝐵 = 𝐷 → ((𝐴 ·o 𝐵) +o 𝐸) = ((𝐴 ·o 𝐷) +o 𝐸))
45	44	adantr 483	. . . . 5 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → ((𝐴 ·o 𝐵) +o 𝐸) = ((𝐴 ·o 𝐷) +o 𝐸))
46	45	eleq2d 2898	. . . 4 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → (((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐸) ↔ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
47	42, 46	mpbidi 243	. . 3 ⊢ ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
48	29, 20, 31, 47	syl3anc 1367	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))
49	39, 48	jaod 855	1 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐵 ∈ 𝐷 ∨ (𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐷) +o 𝐸)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016   ⊆ wss 3936  ∅c0 4291  Ord word 6185  Oncon0 6186  suc csuc 6188  (class class class)co 7150   +_o coa 8093   ·_o comu 8094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-oadd 8100  df-omul 8101

This theorem is referenced by:  omopth2  8204