Theorem omeulem2 8582

Description: Lemma for omeu 8584: 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 1201	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐷 ∈ On)
2		eloni 6374	. . . . . 6 ⊢ (𝐷 ∈ On → Ord 𝐷)
3		ordsucss 7805	. . . . . 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 1199	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐵 ∈ On)
6		onsuc 7798	. . . . . . 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 1197	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐴 ∈ On)
9		simp1r 1198	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐴 ≠ ∅)
10		on0eln0 6420	. . . . . . . 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 256	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ∅ ∈ 𝐴)
13		omword 8569	. . . . . 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 1373	. . . . 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 238	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → (𝐵 ∈ 𝐷 → (𝐴 ·o suc 𝐵) ⊆ (𝐴 ·o 𝐷)))
16		omcl 8535	. . . . . 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 1202	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐸 ∈ 𝐴)
19		onelon 6389	. . . . . 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 8551	. . . . . 6 ⊢ (((𝐴 ·o 𝐷) ∈ On ∧ 𝐸 ∈ On) → (𝐴 ·o 𝐷) ⊆ ((𝐴 ·o 𝐷) +o 𝐸))
22		sstr 3990	. . . . . . 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 414	. . . . . 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 1200	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
28		onelon 6389	. . . . . 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 8535	. . . . . 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 8546	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (𝐶 ∈ 𝐴 ↔ ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴)))
33	32	biimpa 477	. . . . 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 1373	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ ((𝐴 ·o 𝐵) +o 𝐴))
35		omsuc 8525	. . . . 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 2836	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ 𝐴)) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ (𝐴 ·o suc 𝐵))
38		ssel 3975	. . 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 485	. . . . 5 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → 𝐶 ∈ 𝐸)
41		oaord 8546	. . . . 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 243	. . . 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 7416	. . . . . . 7 ⊢ (𝐵 = 𝐷 → (𝐴 ·o 𝐵) = (𝐴 ·o 𝐷))
44	43	oveq1d 7423	. . . . . 6 ⊢ (𝐵 = 𝐷 → ((𝐴 ·o 𝐵) +o 𝐸) = ((𝐴 ·o 𝐷) +o 𝐸))
45	44	adantr 481	. . . . 5 ⊢ ((𝐵 = 𝐷 ∧ 𝐶 ∈ 𝐸) → ((𝐴 ·o 𝐵) +o 𝐸) = ((𝐴 ·o 𝐷) +o 𝐸))
46	45	eleq2d 2819	. . . 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 240	. . 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 1371	. 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 857	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 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940   ⊆ wss 3948  ∅c0 4322  Ord word 6363  Oncon0 6364  suc csuc 6366  (class class class)co 7408   +_o coa 8462   ·_o comu 8463

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-oadd 8469  df-omul 8470

This theorem is referenced by:  omopth2  8583