Theorem shjcom 28217

Description: Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
shjcom	|- ( ( A e. SH /\ B e. SH ) -> ( A vH B ) = ( B vH A ) )

Proof of Theorem shjcom

Step	Hyp	Ref	Expression
1		shjval 28210	. 2 |- ( ( A e. SH /\ B e. SH ) -> ( A vH B ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
2		shjval 28210	. . . 4 |- ( ( B e. SH /\ A e. SH ) -> ( B vH A ) = ( _|_ ` ( _|_ ` ( B u. A ) ) ) )
3	2	ancoms 469	. . 3 |- ( ( A e. SH /\ B e. SH ) -> ( B vH A ) = ( _|_ ` ( _|_ ` ( B u. A ) ) ) )
4		uncom 3757	. . . . 5 |- ( B u. A ) = ( A u. B )
5	4	fveq2i 6194	. . . 4 |- ( _|_ ` ( B u. A ) ) = ( _|_ ` ( A u. B ) )
6	5	fveq2i 6194	. . 3 |- ( _|_ ` ( _|_ ` ( B u. A ) ) ) = ( _|_ ` ( _|_ ` ( A u. B ) ) )
7	3, 6	syl6eq 2672	. 2 |- ( ( A e. SH /\ B e. SH ) -> ( B vH A ) = ( _|_ ` ( _|_ ` ( A u. B ) ) ) )
8	1, 7	eqtr4d 2659	1 |- ( ( A e. SH /\ B e. SH ) -> ( A vH B ) = ( B vH A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   ` cfv 5888  (class class class)co 6650   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-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

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-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-br 4654  df-opab 4713  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sh 28064  df-chj 28169

This theorem is referenced by:  shlej2  28220  shjcomi  28230  shub2  28242  chjcom  28365