Theorem shlej1 28219

Description: Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Assertion

Ref	Expression
shlej1	|- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> ( A vH C ) C_ ( B vH C ) )

Proof of Theorem shlej1

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> A C_ B )
2		unss1 3782	. . . 4 |- ( A C_ B -> ( A u. C ) C_ ( B u. C ) )
3		simpl1 1064	. . . . . . 7 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> A e. SH )
4		shss 28067	. . . . . . 7 |- ( A e. SH -> A C_ ~H )
5	3, 4	syl 17	. . . . . 6 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> A C_ ~H )
6		simpl3 1066	. . . . . . 7 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> C e. SH )
7		shss 28067	. . . . . . 7 |- ( C e. SH -> C C_ ~H )
8	6, 7	syl 17	. . . . . 6 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> C C_ ~H )
9	5, 8	unssd 3789	. . . . 5 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> ( A u. C ) C_ ~H )
10		simpl2 1065	. . . . . . 7 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> B e. SH )
11		shss 28067	. . . . . . 7 |- ( B e. SH -> B C_ ~H )
12	10, 11	syl 17	. . . . . 6 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> B C_ ~H )
13	12, 8	unssd 3789	. . . . 5 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> ( B u. C ) C_ ~H )
14		occon2 28147	. . . . 5 |- ( ( ( A u. C ) C_ ~H /\ ( B u. C ) C_ ~H ) -> ( ( A u. C ) C_ ( B u. C ) -> ( _|_ ` ( _|_ ` ( A u. C ) ) ) C_ ( _|_ ` ( _|_ ` ( B u. C ) ) ) ) )
15	9, 13, 14	syl2anc 693	. . . 4 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> ( ( A u. C ) C_ ( B u. C ) -> ( _|_ ` ( _|_ ` ( A u. C ) ) ) C_ ( _|_ ` ( _|_ ` ( B u. C ) ) ) ) )
16	2, 15	syl5 34	. . 3 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> ( A C_ B -> ( _|_ ` ( _|_ ` ( A u. C ) ) ) C_ ( _|_ ` ( _|_ ` ( B u. C ) ) ) ) )
17	1, 16	mpd 15	. 2 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> ( _|_ ` ( _|_ ` ( A u. C ) ) ) C_ ( _|_ ` ( _|_ ` ( B u. C ) ) ) )
18		shjval 28210	. . 3 |- ( ( A e. SH /\ C e. SH ) -> ( A vH C ) = ( _|_ ` ( _|_ ` ( A u. C ) ) ) )
19	3, 6, 18	syl2anc 693	. 2 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> ( A vH C ) = ( _|_ ` ( _|_ ` ( A u. C ) ) ) )
20		shjval 28210	. . 3 |- ( ( B e. SH /\ C e. SH ) -> ( B vH C ) = ( _|_ ` ( _|_ ` ( B u. C ) ) ) )
21	10, 6, 20	syl2anc 693	. 2 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> ( B vH C ) = ( _|_ ` ( _|_ ` ( B u. C ) ) ) )
22	17, 19, 21	3sstr4d 3648	1 |- ( ( ( A e. SH /\ B e. SH /\ C e. SH ) /\ A C_ B ) -> ( A vH C ) C_ ( B vH C ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   ` cfv 5888  (class class class)co 6650   ~Hchil 27776   SHcsh 27785   _|_cort 27787   vH chj 27790

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-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-hv0cl 27860  ax-hfvmul 27862  ax-hvmul0 27867  ax-hfi 27936  ax-his2 27940  ax-his3 27941

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-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-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-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-sh 28064  df-oc 28109  df-chj 28169

This theorem is referenced by:  shlej2  28220  shlej1i  28237  chlej1  28369