Theorem omeulem2 7663

Description: Lemma for omeu 7665: uniqueness part. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
omeulem2	|- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( ( B e. D \/ ( B = D /\ C e. E ) ) -> ( ( A .o B ) +o C ) e. ( ( A .o D ) +o E ) ) )

Proof of Theorem omeulem2

Step	Hyp	Ref	Expression
1		simp3l 1089	. . . . . 6 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> D e. On )
2		eloni 5733	. . . . . 6 |- ( D e. On -> Ord D )
3		ordsucss 7018	. . . . . 6 |- ( Ord D -> ( B e. D -> suc B C_ D ) )
4	1, 2, 3	3syl 18	. . . . 5 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( B e. D -> suc B C_ D ) )
5		simp2l 1087	. . . . . . 7 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> B e. On )
6		suceloni 7013	. . . . . . 7 |- ( B e. On -> suc B e. On )
7	5, 6	syl 17	. . . . . 6 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> suc B e. On )
8		simp1l 1085	. . . . . 6 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> A e. On )
9		simp1r 1086	. . . . . . 7 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> A =/= (/) )
10		on0eln0 5780	. . . . . . . 8 |- ( A e. On -> ( (/) e. A <-> A =/= (/) ) )
11	8, 10	syl 17	. . . . . . 7 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( (/) e. A <-> A =/= (/) ) )
12	9, 11	mpbird 247	. . . . . 6 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> (/) e. A )
13		omword 7650	. . . . . 6 |- ( ( ( suc B e. On /\ D e. On /\ A e. On ) /\ (/) e. A ) -> ( suc B C_ D <-> ( A .o suc B ) C_ ( A .o D ) ) )
14	7, 1, 8, 12, 13	syl31anc 1329	. . . . 5 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( suc B C_ D <-> ( A .o suc B ) C_ ( A .o D ) ) )
15	4, 14	sylibd 229	. . . 4 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( B e. D -> ( A .o suc B ) C_ ( A .o D ) ) )
16		omcl 7616	. . . . . 6 |- ( ( A e. On /\ D e. On ) -> ( A .o D ) e. On )
17	8, 1, 16	syl2anc 693	. . . . 5 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( A .o D ) e. On )
18		simp3r 1090	. . . . . 6 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> E e. A )
19		onelon 5748	. . . . . 6 |- ( ( A e. On /\ E e. A ) -> E e. On )
20	8, 18, 19	syl2anc 693	. . . . 5 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> E e. On )
21		oaword1 7632	. . . . . 6 |- ( ( ( A .o D ) e. On /\ E e. On ) -> ( A .o D ) C_ ( ( A .o D ) +o E ) )
22		sstr 3611	. . . . . . 7 |- ( ( ( A .o suc B ) C_ ( A .o D ) /\ ( A .o D ) C_ ( ( A .o D ) +o E ) ) -> ( A .o suc B ) C_ ( ( A .o D ) +o E ) )
23	22	expcom 451	. . . . . 6 |- ( ( A .o D ) C_ ( ( A .o D ) +o E ) -> ( ( A .o suc B ) C_ ( A .o D ) -> ( A .o suc B ) C_ ( ( A .o D ) +o E ) ) )
24	21, 23	syl 17	. . . . 5 |- ( ( ( A .o D ) e. On /\ E e. On ) -> ( ( A .o suc B ) C_ ( A .o D ) -> ( A .o suc B ) C_ ( ( A .o D ) +o E ) ) )
25	17, 20, 24	syl2anc 693	. . . 4 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( ( A .o suc B ) C_ ( A .o D ) -> ( A .o suc B ) C_ ( ( A .o D ) +o E ) ) )
26	15, 25	syld 47	. . 3 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( B e. D -> ( A .o suc B ) C_ ( ( A .o D ) +o E ) ) )
27		simp2r 1088	. . . . . 6 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> C e. A )
28		onelon 5748	. . . . . 6 |- ( ( A e. On /\ C e. A ) -> C e. On )
29	8, 27, 28	syl2anc 693	. . . . 5 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> C e. On )
30		omcl 7616	. . . . . 6 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On )
31	8, 5, 30	syl2anc 693	. . . . 5 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( A .o B ) e. On )
32		oaord 7627	. . . . . 6 |- ( ( C e. On /\ A e. On /\ ( A .o B ) e. On ) -> ( C e. A <-> ( ( A .o B ) +o C ) e. ( ( A .o B ) +o A ) ) )
33	32	biimpa 501	. . . . 5 |- ( ( ( C e. On /\ A e. On /\ ( A .o B ) e. On ) /\ C e. A ) -> ( ( A .o B ) +o C ) e. ( ( A .o B ) +o A ) )
34	29, 8, 31, 27, 33	syl31anc 1329	. . . 4 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( ( A .o B ) +o C ) e. ( ( A .o B ) +o A ) )
35		omsuc 7606	. . . . 5 |- ( ( A e. On /\ B e. On ) -> ( A .o suc B ) = ( ( A .o B ) +o A ) )
36	8, 5, 35	syl2anc 693	. . . 4 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( A .o suc B ) = ( ( A .o B ) +o A ) )
37	34, 36	eleqtrrd 2704	. . 3 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( ( A .o B ) +o C ) e. ( A .o suc B ) )
38		ssel 3597	. . 3 |- ( ( A .o suc B ) C_ ( ( A .o D ) +o E ) -> ( ( ( A .o B ) +o C ) e. ( A .o suc B ) -> ( ( A .o B ) +o C ) e. ( ( A .o D ) +o E ) ) )
39	26, 37, 38	syl6ci 71	. 2 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( B e. D -> ( ( A .o B ) +o C ) e. ( ( A .o D ) +o E ) ) )
40		simpr 477	. . . . 5 |- ( ( B = D /\ C e. E ) -> C e. E )
41		oaord 7627	. . . . 5 |- ( ( C e. On /\ E e. On /\ ( A .o B ) e. On ) -> ( C e. E <-> ( ( A .o B ) +o C ) e. ( ( A .o B ) +o E ) ) )
42	40, 41	syl5ib 234	. . . 4 |- ( ( C e. On /\ E e. On /\ ( A .o B ) e. On ) -> ( ( B = D /\ C e. E ) -> ( ( A .o B ) +o C ) e. ( ( A .o B ) +o E ) ) )
43		oveq2 6658	. . . . . . 7 |- ( B = D -> ( A .o B ) = ( A .o D ) )
44	43	oveq1d 6665	. . . . . 6 |- ( B = D -> ( ( A .o B ) +o E ) = ( ( A .o D ) +o E ) )
45	44	adantr 481	. . . . 5 |- ( ( B = D /\ C e. E ) -> ( ( A .o B ) +o E ) = ( ( A .o D ) +o E ) )
46	45	eleq2d 2687	. . . 4 |- ( ( B = D /\ C e. E ) -> ( ( ( A .o B ) +o C ) e. ( ( A .o B ) +o E ) <-> ( ( A .o B ) +o C ) e. ( ( A .o D ) +o E ) ) )
47	42, 46	mpbidi 231	. . 3 |- ( ( C e. On /\ E e. On /\ ( A .o B ) e. On ) -> ( ( B = D /\ C e. E ) -> ( ( A .o B ) +o C ) e. ( ( A .o D ) +o E ) ) )
48	29, 20, 31, 47	syl3anc 1326	. 2 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( ( B = D /\ C e. E ) -> ( ( A .o B ) +o C ) e. ( ( A .o D ) +o E ) ) )
49	39, 48	jaod 395	1 |- ( ( ( A e. On /\ A =/= (/) ) /\ ( B e. On /\ C e. A ) /\ ( D e. On /\ E e. A ) ) -> ( ( B e. D \/ ( B = D /\ C e. E ) ) -> ( ( A .o B ) +o C ) e. ( ( A .o D ) +o E ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   Ord word 5722   Oncon0 5723   suc csuc 5725  (class class class)co 6650   +o coa 7557   .o comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-omul 7565

This theorem is referenced by:  omopth2  7664