Theorem 3oalem2 28522

Description: Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
3oalem1.1	|- B e. CH
3oalem1.2	|- C e. CH
3oalem1.3	|- R e. CH
3oalem1.4	|- S e. CH

Assertion

Ref	Expression
3oalem2	|- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> v e. ( B +H ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) ) )

Distinct variable groups:   x, y, z, w, v, B   x, C, y, z, w, v   x, R, y, z, w, v   x, S, y, z, w, v

Proof of Theorem 3oalem2

Step	Hyp	Ref	Expression
1		simplll 798	. . 3 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> x e. B )
2		simpllr 799	. . . 4 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> y e. R )
3		3oalem1.1	. . . . . . 7 |- B e. CH
4		3oalem1.2	. . . . . . 7 |- C e. CH
5		3oalem1.3	. . . . . . 7 |- R e. CH
6		3oalem1.4	. . . . . . 7 |- S e. CH
7	3, 4, 5, 6	3oalem1 28521	. . . . . 6 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( ( ( x e. ~H /\ y e. ~H ) /\ v e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) )
8		hvaddsub12 27895	. . . . . . . . . 10 |- ( ( y e. ~H /\ w e. ~H /\ w e. ~H ) -> ( y +h ( w -h w ) ) = ( w +h ( y -h w ) ) )
9	8	3anidm23 1385	. . . . . . . . 9 |- ( ( y e. ~H /\ w e. ~H ) -> ( y +h ( w -h w ) ) = ( w +h ( y -h w ) ) )
10		hvsubid 27883	. . . . . . . . . . 11 |- ( w e. ~H -> ( w -h w ) = 0h )
11	10	oveq2d 6666	. . . . . . . . . 10 |- ( w e. ~H -> ( y +h ( w -h w ) ) = ( y +h 0h ) )
12		ax-hvaddid 27861	. . . . . . . . . 10 |- ( y e. ~H -> ( y +h 0h ) = y )
13	11, 12	sylan9eqr 2678	. . . . . . . . 9 |- ( ( y e. ~H /\ w e. ~H ) -> ( y +h ( w -h w ) ) = y )
14	9, 13	eqtr3d 2658	. . . . . . . 8 |- ( ( y e. ~H /\ w e. ~H ) -> ( w +h ( y -h w ) ) = y )
15	14	ad2ant2l 782	. . . . . . 7 |- ( ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) -> ( w +h ( y -h w ) ) = y )
16	15	adantlr 751	. . . . . 6 |- ( ( ( ( x e. ~H /\ y e. ~H ) /\ v e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) -> ( w +h ( y -h w ) ) = y )
17	7, 16	syl 17	. . . . 5 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( w +h ( y -h w ) ) = y )
18		simprlr 803	. . . . . 6 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> w e. S )
19		eqtr2 2642	. . . . . . . . . . 11 |- ( ( v = ( x +h y ) /\ v = ( z +h w ) ) -> ( x +h y ) = ( z +h w ) )
20	19	oveq1d 6665	. . . . . . . . . 10 |- ( ( v = ( x +h y ) /\ v = ( z +h w ) ) -> ( ( x +h y ) -h ( x +h w ) ) = ( ( z +h w ) -h ( x +h w ) ) )
21	20	ad2ant2l 782	. . . . . . . . 9 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( ( x +h y ) -h ( x +h w ) ) = ( ( z +h w ) -h ( x +h w ) ) )
22		simpl 473	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ y e. ~H ) -> x e. ~H )
23	22	anim1i 592	. . . . . . . . . . . . 13 |- ( ( ( x e. ~H /\ y e. ~H ) /\ w e. ~H ) -> ( x e. ~H /\ w e. ~H ) )
24		hvsub4 27894	. . . . . . . . . . . . 13 |- ( ( ( x e. ~H /\ y e. ~H ) /\ ( x e. ~H /\ w e. ~H ) ) -> ( ( x +h y ) -h ( x +h w ) ) = ( ( x -h x ) +h ( y -h w ) ) )
25	23, 24	syldan 487	. . . . . . . . . . . 12 |- ( ( ( x e. ~H /\ y e. ~H ) /\ w e. ~H ) -> ( ( x +h y ) -h ( x +h w ) ) = ( ( x -h x ) +h ( y -h w ) ) )
26		hvsubid 27883	. . . . . . . . . . . . . 14 |- ( x e. ~H -> ( x -h x ) = 0h )
27	26	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( x e. ~H /\ y e. ~H ) /\ w e. ~H ) -> ( x -h x ) = 0h )
28	27	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( x e. ~H /\ y e. ~H ) /\ w e. ~H ) -> ( ( x -h x ) +h ( y -h w ) ) = ( 0h +h ( y -h w ) ) )
29		hvsubcl 27874	. . . . . . . . . . . . . 14 |- ( ( y e. ~H /\ w e. ~H ) -> ( y -h w ) e. ~H )
30		hvaddid2 27880	. . . . . . . . . . . . . 14 |- ( ( y -h w ) e. ~H -> ( 0h +h ( y -h w ) ) = ( y -h w ) )
31	29, 30	syl 17	. . . . . . . . . . . . 13 |- ( ( y e. ~H /\ w e. ~H ) -> ( 0h +h ( y -h w ) ) = ( y -h w ) )
32	31	adantll 750	. . . . . . . . . . . 12 |- ( ( ( x e. ~H /\ y e. ~H ) /\ w e. ~H ) -> ( 0h +h ( y -h w ) ) = ( y -h w ) )
33	25, 28, 32	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( ( x e. ~H /\ y e. ~H ) /\ w e. ~H ) -> ( ( x +h y ) -h ( x +h w ) ) = ( y -h w ) )
34	33	ad2ant2rl 785	. . . . . . . . . 10 |- ( ( ( ( x e. ~H /\ y e. ~H ) /\ v e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( x +h y ) -h ( x +h w ) ) = ( y -h w ) )
35	7, 34	syl 17	. . . . . . . . 9 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( ( x +h y ) -h ( x +h w ) ) = ( y -h w ) )
36		simpr 477	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( z e. ~H /\ w e. ~H ) )
37		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( z e. ~H /\ w e. ~H ) -> w e. ~H )
38	37	anim2i 593	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( x e. ~H /\ w e. ~H ) )
39		hvsub4 27894	. . . . . . . . . . . . . 14 |- ( ( ( z e. ~H /\ w e. ~H ) /\ ( x e. ~H /\ w e. ~H ) ) -> ( ( z +h w ) -h ( x +h w ) ) = ( ( z -h x ) +h ( w -h w ) ) )
40	36, 38, 39	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( x e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( z +h w ) -h ( x +h w ) ) = ( ( z -h x ) +h ( w -h w ) ) )
41	10	ad2antll 765	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( w -h w ) = 0h )
42	41	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( x e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( z -h x ) +h ( w -h w ) ) = ( ( z -h x ) +h 0h ) )
43		hvsubcl 27874	. . . . . . . . . . . . . . . 16 |- ( ( z e. ~H /\ x e. ~H ) -> ( z -h x ) e. ~H )
44		ax-hvaddid 27861	. . . . . . . . . . . . . . . 16 |- ( ( z -h x ) e. ~H -> ( ( z -h x ) +h 0h ) = ( z -h x ) )
45	43, 44	syl 17	. . . . . . . . . . . . . . 15 |- ( ( z e. ~H /\ x e. ~H ) -> ( ( z -h x ) +h 0h ) = ( z -h x ) )
46	45	ancoms 469	. . . . . . . . . . . . . 14 |- ( ( x e. ~H /\ z e. ~H ) -> ( ( z -h x ) +h 0h ) = ( z -h x ) )
47	46	adantrr 753	. . . . . . . . . . . . 13 |- ( ( x e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( z -h x ) +h 0h ) = ( z -h x ) )
48	40, 42, 47	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( x e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( z +h w ) -h ( x +h w ) ) = ( z -h x ) )
49	48	adantlr 751	. . . . . . . . . . 11 |- ( ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( z +h w ) -h ( x +h w ) ) = ( z -h x ) )
50	49	adantlr 751	. . . . . . . . . 10 |- ( ( ( ( x e. ~H /\ y e. ~H ) /\ v e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( z +h w ) -h ( x +h w ) ) = ( z -h x ) )
51	7, 50	syl 17	. . . . . . . . 9 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( ( z +h w ) -h ( x +h w ) ) = ( z -h x ) )
52	21, 35, 51	3eqtr3d 2664	. . . . . . . 8 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( y -h w ) = ( z -h x ) )
53		simpll 790	. . . . . . . . 9 |- ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) -> x e. B )
54		simpll 790	. . . . . . . . 9 |- ( ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) -> z e. C )
55	4	chshii 28084	. . . . . . . . . . . 12 |- C e. SH
56	3	chshii 28084	. . . . . . . . . . . 12 |- B e. SH
57	55, 56	shsvsi 28226	. . . . . . . . . . 11 |- ( ( z e. C /\ x e. B ) -> ( z -h x ) e. ( C +H B ) )
58	57	ancoms 469	. . . . . . . . . 10 |- ( ( x e. B /\ z e. C ) -> ( z -h x ) e. ( C +H B ) )
59	56, 55	shscomi 28222	. . . . . . . . . 10 |- ( B +H C ) = ( C +H B )
60	58, 59	syl6eleqr 2712	. . . . . . . . 9 |- ( ( x e. B /\ z e. C ) -> ( z -h x ) e. ( B +H C ) )
61	53, 54, 60	syl2an 494	. . . . . . . 8 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( z -h x ) e. ( B +H C ) )
62	52, 61	eqeltrd 2701	. . . . . . 7 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( y -h w ) e. ( B +H C ) )
63		simplr 792	. . . . . . . 8 |- ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) -> y e. R )
64		simplr 792	. . . . . . . 8 |- ( ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) -> w e. S )
65	5	chshii 28084	. . . . . . . . 9 |- R e. SH
66	6	chshii 28084	. . . . . . . . 9 |- S e. SH
67	65, 66	shsvsi 28226	. . . . . . . 8 |- ( ( y e. R /\ w e. S ) -> ( y -h w ) e. ( R +H S ) )
68	63, 64, 67	syl2an 494	. . . . . . 7 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( y -h w ) e. ( R +H S ) )
69	62, 68	elind 3798	. . . . . 6 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( y -h w ) e. ( ( B +H C ) i^i ( R +H S ) ) )
70	56, 55	shscli 28176	. . . . . . . 8 |- ( B +H C ) e. SH
71	65, 66	shscli 28176	. . . . . . . 8 |- ( R +H S ) e. SH
72	70, 71	shincli 28221	. . . . . . 7 |- ( ( B +H C ) i^i ( R +H S ) ) e. SH
73	66, 72	shsvai 28223	. . . . . 6 |- ( ( w e. S /\ ( y -h w ) e. ( ( B +H C ) i^i ( R +H S ) ) ) -> ( w +h ( y -h w ) ) e. ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) )
74	18, 69, 73	syl2anc 693	. . . . 5 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( w +h ( y -h w ) ) e. ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) )
75	17, 74	eqeltrrd 2702	. . . 4 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> y e. ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) )
76	2, 75	elind 3798	. . 3 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> y e. ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) )
77	66, 72	shscli 28176	. . . . 5 |- ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) e. SH
78	65, 77	shincli 28221	. . . 4 |- ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) e. SH
79	56, 78	shsvai 28223	. . 3 |- ( ( x e. B /\ y e. ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) ) -> ( x +h y ) e. ( B +H ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) ) )
80	1, 76, 79	syl2anc 693	. 2 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( x +h y ) e. ( B +H ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) ) )
81		eleq1 2689	. . 3 |- ( v = ( x +h y ) -> ( v e. ( B +H ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) ) <-> ( x +h y ) e. ( B +H ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) ) ) )
82	81	ad2antlr 763	. 2 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> ( v e. ( B +H ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) ) <-> ( x +h y ) e. ( B +H ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) ) ) )
83	80, 82	mpbird 247	1 |- ( ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) /\ ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) ) -> v e. ( B +H ( R i^i ( S +H ( ( B +H C ) i^i ( R +H S ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573  (class class class)co 6650   ~Hchil 27776   +h cva 27777   0hc0v 27781   -h cmv 27782   CHcch 27786   +H cph 27788

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867

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-nel 2898  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-int 4476  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269  df-nn 11021  df-grpo 27347  df-ablo 27399  df-hvsub 27828  df-hlim 27829  df-sh 28064  df-ch 28078  df-shs 28167

This theorem is referenced by:  3oalem3  28523