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Theorem shjcom 27601
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 27594 . 2 ((𝐴S𝐵S ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
2 shjval 27594 . . . 4 ((𝐵S𝐴S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐵𝐴))))
32ancoms 468 . . 3 ((𝐴S𝐵S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐵𝐴))))
4 uncom 3719 . . . . 5 (𝐵𝐴) = (𝐴𝐵)
54fveq2i 6106 . . . 4 (⊥‘(𝐵𝐴)) = (⊥‘(𝐴𝐵))
65fveq2i 6106 . . 3 (⊥‘(⊥‘(𝐵𝐴))) = (⊥‘(⊥‘(𝐴𝐵)))
73, 6syl6eq 2660 . 2 ((𝐴S𝐵S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐴𝐵))))
81, 7eqtr4d 2647 1 ((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cun 3538  cfv 5804  (class class class)co 6549   S csh 27169  cort 27171   chj 27174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-hilex 27240
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-sh 27448  df-chj 27553
This theorem is referenced by:  shlej2  27604  shjcomi  27614  shub2  27626  chjcom  27749
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