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Theorem shjcom 30469
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 30462 . 2 ((𝐴S𝐵S ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
2 shjval 30462 . . . 4 ((𝐵S𝐴S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐵𝐴))))
32ancoms 459 . . 3 ((𝐴S𝐵S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐵𝐴))))
4 uncom 4146 . . . . 5 (𝐵𝐴) = (𝐴𝐵)
54fveq2i 6878 . . . 4 (⊥‘(𝐵𝐴)) = (⊥‘(𝐴𝐵))
65fveq2i 6878 . . 3 (⊥‘(⊥‘(𝐵𝐴))) = (⊥‘(⊥‘(𝐴𝐵)))
73, 6eqtrdi 2787 . 2 ((𝐴S𝐵S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐴𝐵))))
81, 7eqtr4d 2774 1 ((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cun 3939  cfv 6529  (class class class)co 7390   S csh 30039  cort 30041   chj 30044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-hilex 30110
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6481  df-fun 6531  df-fv 6537  df-ov 7393  df-oprab 7394  df-mpo 7395  df-sh 30318  df-chj 30421
This theorem is referenced by:  shlej2  30472  shjcomi  30482  shub2  30494  chjcom  30617
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