Theorem 3oalem1 28521

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
3oalem1	|- ( ( ( ( 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 ) ) )

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 3oalem1

Step	Hyp	Ref	Expression
1		3oalem1.1	. . . . 5 |- B e. CH
2	1	cheli 28089	. . . 4 |- ( x e. B -> x e. ~H )
3		3oalem1.3	. . . . 5 |- R e. CH
4	3	cheli 28089	. . . 4 |- ( y e. R -> y e. ~H )
5	2, 4	anim12i 590	. . 3 |- ( ( x e. B /\ y e. R ) -> ( x e. ~H /\ y e. ~H ) )
6		hvaddcl 27869	. . . . 5 |- ( ( x e. ~H /\ y e. ~H ) -> ( x +h y ) e. ~H )
7		eleq1 2689	. . . . 5 |- ( v = ( x +h y ) -> ( v e. ~H <-> ( x +h y ) e. ~H ) )
8	6, 7	syl5ibrcom 237	. . . 4 |- ( ( x e. ~H /\ y e. ~H ) -> ( v = ( x +h y ) -> v e. ~H ) )
9	8	imdistani 726	. . 3 |- ( ( ( x e. ~H /\ y e. ~H ) /\ v = ( x +h y ) ) -> ( ( x e. ~H /\ y e. ~H ) /\ v e. ~H ) )
10	5, 9	sylan 488	. 2 |- ( ( ( x e. B /\ y e. R ) /\ v = ( x +h y ) ) -> ( ( x e. ~H /\ y e. ~H ) /\ v e. ~H ) )
11		3oalem1.2	. . . . 5 |- C e. CH
12	11	cheli 28089	. . . 4 |- ( z e. C -> z e. ~H )
13		3oalem1.4	. . . . 5 |- S e. CH
14	13	cheli 28089	. . . 4 |- ( w e. S -> w e. ~H )
15	12, 14	anim12i 590	. . 3 |- ( ( z e. C /\ w e. S ) -> ( z e. ~H /\ w e. ~H ) )
16	15	adantr 481	. 2 |- ( ( ( z e. C /\ w e. S ) /\ v = ( z +h w ) ) -> ( z e. ~H /\ w e. ~H ) )
17	10, 16	anim12i 590	1 |- ( ( ( ( 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990  (class class class)co 6650   ~Hchil 27776   +h cva 27777   CHcch 27786

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-hilex 27856  ax-hfvadd 27857

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-fv 5896  df-ov 6653  df-sh 28064  df-ch 28078

This theorem is referenced by:  3oalem2  28522