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Theorem shjcom 28066
Description: Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
shjcom ((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))

Proof of Theorem shjcom
StepHypRef Expression
1 shjval 28059 . 2 ((𝐴S𝐵S ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
2 shjval 28059 . . . 4 ((𝐵S𝐴S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐵𝐴))))
32ancoms 469 . . 3 ((𝐴S𝐵S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐵𝐴))))
4 uncom 3735 . . . . 5 (𝐵𝐴) = (𝐴𝐵)
54fveq2i 6151 . . . 4 (⊥‘(𝐵𝐴)) = (⊥‘(𝐴𝐵))
65fveq2i 6151 . . 3 (⊥‘(⊥‘(𝐵𝐴))) = (⊥‘(⊥‘(𝐴𝐵)))
73, 6syl6eq 2671 . 2 ((𝐴S𝐵S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐴𝐵))))
81, 7eqtr4d 2658 1 ((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cun 3553  cfv 5847  (class class class)co 6604   S csh 27634  cort 27636   chj 27639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-hilex 27705
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-sh 27913  df-chj 28018
This theorem is referenced by:  shlej2  28069  shjcomi  28079  shub2  28091  chjcom  28214
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