Theorem 5oalem2 28514

Description: Lemma for orthoarguesian law 5OA. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)

Hypotheses

Ref	Expression
5oalem2.1	|- A e. SH
5oalem2.2	|- B e. SH
5oalem2.3	|- C e. SH
5oalem2.4	|- D e. SH

Assertion

Ref	Expression
5oalem2	|- ( ( ( ( x e. A /\ y e. B ) /\ ( z e. C /\ w e. D ) ) /\ ( x +h y ) = ( z +h w ) ) -> ( x -h z ) e. ( ( A +H C ) i^i ( B +H D ) ) )

Proof of Theorem 5oalem2

Step	Hyp	Ref	Expression
1		5oalem2.1	. . . . 5 |- A e. SH
2		5oalem2.3	. . . . 5 |- C e. SH
3	1, 2	shsvsi 28226	. . . 4 |- ( ( x e. A /\ z e. C ) -> ( x -h z ) e. ( A +H C ) )
4	3	ad2ant2r 783	. . 3 |- ( ( ( x e. A /\ y e. B ) /\ ( z e. C /\ w e. D ) ) -> ( x -h z ) e. ( A +H C ) )
5	4	adantr 481	. 2 |- ( ( ( ( x e. A /\ y e. B ) /\ ( z e. C /\ w e. D ) ) /\ ( x +h y ) = ( z +h w ) ) -> ( x -h z ) e. ( A +H C ) )
6		5oalem2.4	. . . . . . . 8 |- D e. SH
7		5oalem2.2	. . . . . . . 8 |- B e. SH
8	6, 7	shsvsi 28226	. . . . . . 7 |- ( ( w e. D /\ y e. B ) -> ( w -h y ) e. ( D +H B ) )
9	8	ancoms 469	. . . . . 6 |- ( ( y e. B /\ w e. D ) -> ( w -h y ) e. ( D +H B ) )
10	7, 6	shscomi 28222	. . . . . 6 |- ( B +H D ) = ( D +H B )
11	9, 10	syl6eleqr 2712	. . . . 5 |- ( ( y e. B /\ w e. D ) -> ( w -h y ) e. ( B +H D ) )
12	11	ad2ant2l 782	. . . 4 |- ( ( ( x e. A /\ y e. B ) /\ ( z e. C /\ w e. D ) ) -> ( w -h y ) e. ( B +H D ) )
13	12	adantr 481	. . 3 |- ( ( ( ( x e. A /\ y e. B ) /\ ( z e. C /\ w e. D ) ) /\ ( x +h y ) = ( z +h w ) ) -> ( w -h y ) e. ( B +H D ) )
14	1	sheli 28071	. . . . . 6 |- ( x e. A -> x e. ~H )
15	7	sheli 28071	. . . . . 6 |- ( y e. B -> y e. ~H )
16	14, 15	anim12i 590	. . . . 5 |- ( ( x e. A /\ y e. B ) -> ( x e. ~H /\ y e. ~H ) )
17	2	sheli 28071	. . . . . 6 |- ( z e. C -> z e. ~H )
18	6	sheli 28071	. . . . . 6 |- ( w e. D -> w e. ~H )
19	17, 18	anim12i 590	. . . . 5 |- ( ( z e. C /\ w e. D ) -> ( z e. ~H /\ w e. ~H ) )
20	16, 19	anim12i 590	. . . 4 |- ( ( ( x e. A /\ y e. B ) /\ ( z e. C /\ w e. D ) ) -> ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) )
21		oveq1 6657	. . . . . . 7 |- ( ( x +h y ) = ( z +h w ) -> ( ( x +h y ) -h ( z +h y ) ) = ( ( z +h w ) -h ( z +h y ) ) )
22	21	adantl 482	. . . . . 6 |- ( ( ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) /\ ( x +h y ) = ( z +h w ) ) -> ( ( x +h y ) -h ( z +h y ) ) = ( ( z +h w ) -h ( z +h y ) ) )
23		simpr 477	. . . . . . . . . . . 12 |- ( ( x e. ~H /\ y e. ~H ) -> y e. ~H )
24	23	anim2i 593	. . . . . . . . . . 11 |- ( ( z e. ~H /\ ( x e. ~H /\ y e. ~H ) ) -> ( z e. ~H /\ y e. ~H ) )
25	24	ancoms 469	. . . . . . . . . 10 |- ( ( ( x e. ~H /\ y e. ~H ) /\ z e. ~H ) -> ( z e. ~H /\ y e. ~H ) )
26		hvsub4 27894	. . . . . . . . . 10 |- ( ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ y e. ~H ) ) -> ( ( x +h y ) -h ( z +h y ) ) = ( ( x -h z ) +h ( y -h y ) ) )
27	25, 26	syldan 487	. . . . . . . . 9 |- ( ( ( x e. ~H /\ y e. ~H ) /\ z e. ~H ) -> ( ( x +h y ) -h ( z +h y ) ) = ( ( x -h z ) +h ( y -h y ) ) )
28		hvsubid 27883	. . . . . . . . . . 11 |- ( y e. ~H -> ( y -h y ) = 0h )
29	28	oveq2d 6666	. . . . . . . . . 10 |- ( y e. ~H -> ( ( x -h z ) +h ( y -h y ) ) = ( ( x -h z ) +h 0h ) )
30	29	ad2antlr 763	. . . . . . . . 9 |- ( ( ( x e. ~H /\ y e. ~H ) /\ z e. ~H ) -> ( ( x -h z ) +h ( y -h y ) ) = ( ( x -h z ) +h 0h ) )
31		hvsubcl 27874	. . . . . . . . . . 11 |- ( ( x e. ~H /\ z e. ~H ) -> ( x -h z ) e. ~H )
32		ax-hvaddid 27861	. . . . . . . . . . 11 |- ( ( x -h z ) e. ~H -> ( ( x -h z ) +h 0h ) = ( x -h z ) )
33	31, 32	syl 17	. . . . . . . . . 10 |- ( ( x e. ~H /\ z e. ~H ) -> ( ( x -h z ) +h 0h ) = ( x -h z ) )
34	33	adantlr 751	. . . . . . . . 9 |- ( ( ( x e. ~H /\ y e. ~H ) /\ z e. ~H ) -> ( ( x -h z ) +h 0h ) = ( x -h z ) )
35	27, 30, 34	3eqtrd 2660	. . . . . . . 8 |- ( ( ( x e. ~H /\ y e. ~H ) /\ z e. ~H ) -> ( ( x +h y ) -h ( z +h y ) ) = ( x -h z ) )
36	35	adantrr 753	. . . . . . 7 |- ( ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( x +h y ) -h ( z +h y ) ) = ( x -h z ) )
37	36	adantr 481	. . . . . 6 |- ( ( ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) /\ ( x +h y ) = ( z +h w ) ) -> ( ( x +h y ) -h ( z +h y ) ) = ( x -h z ) )
38		simpr 477	. . . . . . . . . 10 |- ( ( y e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( z e. ~H /\ w e. ~H ) )
39		simpl 473	. . . . . . . . . . . 12 |- ( ( z e. ~H /\ w e. ~H ) -> z e. ~H )
40	39	anim1i 592	. . . . . . . . . . 11 |- ( ( ( z e. ~H /\ w e. ~H ) /\ y e. ~H ) -> ( z e. ~H /\ y e. ~H ) )
41	40	ancoms 469	. . . . . . . . . 10 |- ( ( y e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( z e. ~H /\ y e. ~H ) )
42		hvsub4 27894	. . . . . . . . . 10 |- ( ( ( z e. ~H /\ w e. ~H ) /\ ( z e. ~H /\ y e. ~H ) ) -> ( ( z +h w ) -h ( z +h y ) ) = ( ( z -h z ) +h ( w -h y ) ) )
43	38, 41, 42	syl2anc 693	. . . . . . . . 9 |- ( ( y e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( z +h w ) -h ( z +h y ) ) = ( ( z -h z ) +h ( w -h y ) ) )
44		hvsubid 27883	. . . . . . . . . . 11 |- ( z e. ~H -> ( z -h z ) = 0h )
45	44	oveq1d 6665	. . . . . . . . . 10 |- ( z e. ~H -> ( ( z -h z ) +h ( w -h y ) ) = ( 0h +h ( w -h y ) ) )
46	45	ad2antrl 764	. . . . . . . . 9 |- ( ( y e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( z -h z ) +h ( w -h y ) ) = ( 0h +h ( w -h y ) ) )
47		hvsubcl 27874	. . . . . . . . . . . 12 |- ( ( w e. ~H /\ y e. ~H ) -> ( w -h y ) e. ~H )
48		hvaddid2 27880	. . . . . . . . . . . 12 |- ( ( w -h y ) e. ~H -> ( 0h +h ( w -h y ) ) = ( w -h y ) )
49	47, 48	syl 17	. . . . . . . . . . 11 |- ( ( w e. ~H /\ y e. ~H ) -> ( 0h +h ( w -h y ) ) = ( w -h y ) )
50	49	ancoms 469	. . . . . . . . . 10 |- ( ( y e. ~H /\ w e. ~H ) -> ( 0h +h ( w -h y ) ) = ( w -h y ) )
51	50	adantrl 752	. . . . . . . . 9 |- ( ( y e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( 0h +h ( w -h y ) ) = ( w -h y ) )
52	43, 46, 51	3eqtrd 2660	. . . . . . . 8 |- ( ( y e. ~H /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( z +h w ) -h ( z +h y ) ) = ( w -h y ) )
53	52	adantll 750	. . . . . . 7 |- ( ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) -> ( ( z +h w ) -h ( z +h y ) ) = ( w -h y ) )
54	53	adantr 481	. . . . . 6 |- ( ( ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) /\ ( x +h y ) = ( z +h w ) ) -> ( ( z +h w ) -h ( z +h y ) ) = ( w -h y ) )
55	22, 37, 54	3eqtr3d 2664	. . . . 5 |- ( ( ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) /\ ( x +h y ) = ( z +h w ) ) -> ( x -h z ) = ( w -h y ) )
56	55	eleq1d 2686	. . . 4 |- ( ( ( ( x e. ~H /\ y e. ~H ) /\ ( z e. ~H /\ w e. ~H ) ) /\ ( x +h y ) = ( z +h w ) ) -> ( ( x -h z ) e. ( B +H D ) <-> ( w -h y ) e. ( B +H D ) ) )
57	20, 56	sylan 488	. . 3 |- ( ( ( ( x e. A /\ y e. B ) /\ ( z e. C /\ w e. D ) ) /\ ( x +h y ) = ( z +h w ) ) -> ( ( x -h z ) e. ( B +H D ) <-> ( w -h y ) e. ( B +H D ) ) )
58	13, 57	mpbird 247	. 2 |- ( ( ( ( x e. A /\ y e. B ) /\ ( z e. C /\ w e. D ) ) /\ ( x +h y ) = ( z +h w ) ) -> ( x -h z ) e. ( B +H D ) )
59	5, 58	elind 3798	1 |- ( ( ( ( x e. A /\ y e. B ) /\ ( z e. C /\ w e. D ) ) /\ ( x +h y ) = ( z +h w ) ) -> ( x -h z ) e. ( ( A +H C ) i^i ( B +H D ) ) )

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   SHcsh 27785   +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-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:  5oalem3  28515  5oalem4  28516