Theorem shuni 28159

Description: Two subspaces with trivial intersection have a unique decomposition of the elements of the subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
shuni.1	|- ( ph -> H e. SH )
shuni.2	|- ( ph -> K e. SH )
shuni.3	|- ( ph -> ( H i^i K ) = 0H )
shuni.4	|- ( ph -> A e. H )
shuni.5	|- ( ph -> B e. K )
shuni.6	|- ( ph -> C e. H )
shuni.7	|- ( ph -> D e. K )
shuni.8	|- ( ph -> ( A +h B ) = ( C +h D ) )

Assertion

Ref	Expression
shuni	|- ( ph -> ( A = C /\ B = D ) )

Proof of Theorem shuni

Step	Hyp	Ref	Expression
1		shuni.1	. . . . . . 7 |- ( ph -> H e. SH )
2		shuni.4	. . . . . . 7 |- ( ph -> A e. H )
3		shuni.6	. . . . . . 7 |- ( ph -> C e. H )
4		shsubcl 28077	. . . . . . 7 |- ( ( H e. SH /\ A e. H /\ C e. H ) -> ( A -h C ) e. H )
5	1, 2, 3, 4	syl3anc 1326	. . . . . 6 |- ( ph -> ( A -h C ) e. H )
6		shuni.8	. . . . . . . 8 |- ( ph -> ( A +h B ) = ( C +h D ) )
7		shel 28068	. . . . . . . . . 10 |- ( ( H e. SH /\ A e. H ) -> A e. ~H )
8	1, 2, 7	syl2anc 693	. . . . . . . . 9 |- ( ph -> A e. ~H )
9		shuni.2	. . . . . . . . . 10 |- ( ph -> K e. SH )
10		shuni.5	. . . . . . . . . 10 |- ( ph -> B e. K )
11		shel 28068	. . . . . . . . . 10 |- ( ( K e. SH /\ B e. K ) -> B e. ~H )
12	9, 10, 11	syl2anc 693	. . . . . . . . 9 |- ( ph -> B e. ~H )
13		shel 28068	. . . . . . . . . 10 |- ( ( H e. SH /\ C e. H ) -> C e. ~H )
14	1, 3, 13	syl2anc 693	. . . . . . . . 9 |- ( ph -> C e. ~H )
15		shuni.7	. . . . . . . . . 10 |- ( ph -> D e. K )
16		shel 28068	. . . . . . . . . 10 |- ( ( K e. SH /\ D e. K ) -> D e. ~H )
17	9, 15, 16	syl2anc 693	. . . . . . . . 9 |- ( ph -> D e. ~H )
18		hvaddsub4 27935	. . . . . . . . 9 |- ( ( ( A e. ~H /\ B e. ~H ) /\ ( C e. ~H /\ D e. ~H ) ) -> ( ( A +h B ) = ( C +h D ) <-> ( A -h C ) = ( D -h B ) ) )
19	8, 12, 14, 17, 18	syl22anc 1327	. . . . . . . 8 |- ( ph -> ( ( A +h B ) = ( C +h D ) <-> ( A -h C ) = ( D -h B ) ) )
20	6, 19	mpbid 222	. . . . . . 7 |- ( ph -> ( A -h C ) = ( D -h B ) )
21		shsubcl 28077	. . . . . . . 8 |- ( ( K e. SH /\ D e. K /\ B e. K ) -> ( D -h B ) e. K )
22	9, 15, 10, 21	syl3anc 1326	. . . . . . 7 |- ( ph -> ( D -h B ) e. K )
23	20, 22	eqeltrd 2701	. . . . . 6 |- ( ph -> ( A -h C ) e. K )
24	5, 23	elind 3798	. . . . 5 |- ( ph -> ( A -h C ) e. ( H i^i K ) )
25		shuni.3	. . . . 5 |- ( ph -> ( H i^i K ) = 0H )
26	24, 25	eleqtrd 2703	. . . 4 |- ( ph -> ( A -h C ) e. 0H )
27		elch0 28111	. . . 4 |- ( ( A -h C ) e. 0H <-> ( A -h C ) = 0h )
28	26, 27	sylib 208	. . 3 |- ( ph -> ( A -h C ) = 0h )
29		hvsubeq0 27925	. . . 4 |- ( ( A e. ~H /\ C e. ~H ) -> ( ( A -h C ) = 0h <-> A = C ) )
30	8, 14, 29	syl2anc 693	. . 3 |- ( ph -> ( ( A -h C ) = 0h <-> A = C ) )
31	28, 30	mpbid 222	. 2 |- ( ph -> A = C )
32	20, 28	eqtr3d 2658	. . . 4 |- ( ph -> ( D -h B ) = 0h )
33		hvsubeq0 27925	. . . . 5 |- ( ( D e. ~H /\ B e. ~H ) -> ( ( D -h B ) = 0h <-> D = B ) )
34	17, 12, 33	syl2anc 693	. . . 4 |- ( ph -> ( ( D -h B ) = 0h <-> D = B ) )
35	32, 34	mpbid 222	. . 3 |- ( ph -> D = B )
36	35	eqcomd 2628	. 2 |- ( ph -> B = D )
37	31, 36	jca 554	1 |- ( ph -> ( A = C /\ B = 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   0Hc0h 27792

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-pre-mulgt0 10013  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-hvmulass 27864  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-rmo 2920  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-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-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-hvsub 27828  df-sh 28064  df-ch0 28110

This theorem is referenced by:  chocunii  28160  pjhthmo  28161  chscllem3  28498