Theorem 5oalem1 28513

Description: Lemma for orthoarguesian law 5OA. (Contributed by NM, 1-Apr-2000.) (New usage is discouraged.)

Hypotheses

Ref	Expression
5oalem1.1	|- A e. SH
5oalem1.2	|- B e. SH
5oalem1.3	|- C e. SH
5oalem1.4	|- R e. SH

Assertion

Ref	Expression
5oalem1	|- ( ( ( ( x e. A /\ y e. B ) /\ v = ( x +h y ) ) /\ ( z e. C /\ ( x -h z ) e. R ) ) -> v e. ( B +H ( A i^i ( C +H R ) ) ) )

Proof of Theorem 5oalem1

Step	Hyp	Ref	Expression
1		simplll 798	. . . 4 |- ( ( ( ( x e. A /\ y e. B ) /\ v = ( x +h y ) ) /\ ( z e. C /\ ( x -h z ) e. R ) ) -> x e. A )
2		5oalem1.1	. . . . . . . 8 |- A e. SH
3	2	sheli 28071	. . . . . . 7 |- ( x e. A -> x e. ~H )
4	3	ad2antrr 762	. . . . . 6 |- ( ( ( x e. A /\ y e. B ) /\ v = ( x +h y ) ) -> x e. ~H )
5		5oalem1.3	. . . . . . . 8 |- C e. SH
6	5	sheli 28071	. . . . . . 7 |- ( z e. C -> z e. ~H )
7	6	adantr 481	. . . . . 6 |- ( ( z e. C /\ ( x -h z ) e. R ) -> z e. ~H )
8		hvaddsub12 27895	. . . . . . . 8 |- ( ( x e. ~H /\ z e. ~H /\ z e. ~H ) -> ( x +h ( z -h z ) ) = ( z +h ( x -h z ) ) )
9	8	3anidm23 1385	. . . . . . 7 |- ( ( x e. ~H /\ z e. ~H ) -> ( x +h ( z -h z ) ) = ( z +h ( x -h z ) ) )
10		hvsubid 27883	. . . . . . . . 9 |- ( z e. ~H -> ( z -h z ) = 0h )
11	10	oveq2d 6666	. . . . . . . 8 |- ( z e. ~H -> ( x +h ( z -h z ) ) = ( x +h 0h ) )
12		ax-hvaddid 27861	. . . . . . . 8 |- ( x e. ~H -> ( x +h 0h ) = x )
13	11, 12	sylan9eqr 2678	. . . . . . 7 |- ( ( x e. ~H /\ z e. ~H ) -> ( x +h ( z -h z ) ) = x )
14	9, 13	eqtr3d 2658	. . . . . 6 |- ( ( x e. ~H /\ z e. ~H ) -> ( z +h ( x -h z ) ) = x )
15	4, 7, 14	syl2an 494	. . . . 5 |- ( ( ( ( x e. A /\ y e. B ) /\ v = ( x +h y ) ) /\ ( z e. C /\ ( x -h z ) e. R ) ) -> ( z +h ( x -h z ) ) = x )
16		5oalem1.4	. . . . . . 7 |- R e. SH
17	5, 16	shsvai 28223	. . . . . 6 |- ( ( z e. C /\ ( x -h z ) e. R ) -> ( z +h ( x -h z ) ) e. ( C +H R ) )
18	17	adantl 482	. . . . 5 |- ( ( ( ( x e. A /\ y e. B ) /\ v = ( x +h y ) ) /\ ( z e. C /\ ( x -h z ) e. R ) ) -> ( z +h ( x -h z ) ) e. ( C +H R ) )
19	15, 18	eqeltrrd 2702	. . . 4 |- ( ( ( ( x e. A /\ y e. B ) /\ v = ( x +h y ) ) /\ ( z e. C /\ ( x -h z ) e. R ) ) -> x e. ( C +H R ) )
20	1, 19	elind 3798	. . 3 |- ( ( ( ( x e. A /\ y e. B ) /\ v = ( x +h y ) ) /\ ( z e. C /\ ( x -h z ) e. R ) ) -> x e. ( A i^i ( C +H R ) ) )
21		simpllr 799	. . 3 |- ( ( ( ( x e. A /\ y e. B ) /\ v = ( x +h y ) ) /\ ( z e. C /\ ( x -h z ) e. R ) ) -> y e. B )
22	5, 16	shscli 28176	. . . . . 6 |- ( C +H R ) e. SH
23	2, 22	shincli 28221	. . . . 5 |- ( A i^i ( C +H R ) ) e. SH
24		5oalem1.2	. . . . 5 |- B e. SH
25	23, 24	shsvai 28223	. . . 4 |- ( ( x e. ( A i^i ( C +H R ) ) /\ y e. B ) -> ( x +h y ) e. ( ( A i^i ( C +H R ) ) +H B ) )
26	23, 24	shscomi 28222	. . . 4 |- ( ( A i^i ( C +H R ) ) +H B ) = ( B +H ( A i^i ( C +H R ) ) )
27	25, 26	syl6eleq 2711	. . 3 |- ( ( x e. ( A i^i ( C +H R ) ) /\ y e. B ) -> ( x +h y ) e. ( B +H ( A i^i ( C +H R ) ) ) )
28	20, 21, 27	syl2anc 693	. 2 |- ( ( ( ( x e. A /\ y e. B ) /\ v = ( x +h y ) ) /\ ( z e. C /\ ( x -h z ) e. R ) ) -> ( x +h y ) e. ( B +H ( A i^i ( C +H R ) ) ) )
29		eleq1 2689	. . 3 |- ( v = ( x +h y ) -> ( v e. ( B +H ( A i^i ( C +H R ) ) ) <-> ( x +h y ) e. ( B +H ( A i^i ( C +H R ) ) ) ) )
30	29	ad2antlr 763	. 2 |- ( ( ( ( x e. A /\ y e. B ) /\ v = ( x +h y ) ) /\ ( z e. C /\ ( x -h z ) e. R ) ) -> ( v e. ( B +H ( A i^i ( C +H R ) ) ) <-> ( x +h y ) e. ( B +H ( A i^i ( C +H R ) ) ) ) )
31	28, 30	mpbird 247	1 |- ( ( ( ( x e. A /\ y e. B ) /\ v = ( x +h y ) ) /\ ( z e. C /\ ( x -h z ) e. R ) ) -> v e. ( B +H ( A i^i ( C +H R ) ) ) )

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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-er 7742  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-grpo 27347  df-ablo 27399  df-hvsub 27828  df-sh 28064  df-shs 28167

This theorem is referenced by:  5oalem6  28518