Theorem omeulem2 8498

Description: Lemma for omeu 8500: 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 1202	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐷 ∈ On)
2		eloni 6316	. . . . . 6 ⊢ (𝐷 ∈ On → Ord 𝐷)
3		ordsucss 7748	. . . . . 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 1200	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐵 ∈ On)
6		onsuc 7743	. . . . . . 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 1198	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐴 ∈ On)
9		simp1r 1199	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐴 ≠ ∅)
10		on0eln0 6363	. . . . . . . 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 257	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ∅ ∈ 𝐴)
13		omword 8485	. . . . . 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 1375	. . . . 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 239	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐵 ∈ 𝐷 → (𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷)))
16		omcl 8451	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐷 ∈ On) → (𝐴 ·o 𝐷) ∈ On)
17	8, 1, 16	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐴 ·o 𝐷) ∈ On)
18		simp3r 1203	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐸 ∈ 𝐴)
19		onelon 6331	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐸 ∈ 𝐴) → 𝐸 ∈ On)
20	8, 18, 19	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐸 ∈ On)
21		oaword1 8467	. . . . . 6 ⊢ (((𝐴 ·o 𝐷) ∈ On ∧ 𝐸 ∈ On) → (𝐴 ·o 𝐷) ⊆ ((𝐴 ·o 𝐷) +o 𝐸))
22		sstr 3943	. . . . . . 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 413	. . . . . 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 584	. . . 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 1201	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
28		onelon 6331	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On)
29	8, 27, 28	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐶 ∈ On)
30		omcl 8451	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
31	8, 5, 30	syl2anc 584	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐴 ·o 𝐵) ∈ On)
32		oaord 8462	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (𝐶 ∈ 𝐴 ↔ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴)))
33	32	biimpa 476	. . . . 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 1375	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴))
35		omsuc 8441	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴))
36	8, 5, 35	syl2anc 584	. . . 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 2834	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ (𝐴 ·o suc 𝐵))
38		ssel 3928	. . 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 484	. . . . 5 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → 𝐶 ∈ 𝐸)
41		oaord 8462	. . . . 5 ⊢ ((𝐶 ∈ On ∧ 𝐸 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (𝐶 ∈ 𝐸 ↔ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐸)))
42	40, 41	imbitrid 244	. . . 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 7354	. . . . . . 7 ⊢ (𝐵 = 𝐷 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐷))
44	43	oveq1d 7361	. . . . . 6 ⊢ (𝐵 = 𝐷 → ((𝐴 ·o 𝐵) +o 𝐸) = ((𝐴 ·o 𝐷) +o 𝐸))
45	44	adantr 480	. . . . 5 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → ((𝐴 ·o 𝐵) +o 𝐸) = ((𝐴 ·o 𝐷) +o 𝐸))
46	45	eleq2d 2817	. . . 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 241	. . 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 1373	. 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 859	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   ⊆ wss 3902  ∅c0 4283  Ord word 6305  Oncon0 6306  suc csuc 6308  (class class class)co 7346   +_o coa 8382   ·_o comu 8383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-oadd 8389  df-omul 8390

This theorem is referenced by:  omopth2  8499