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Theorem shjcom 31387
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 31380 . 2 ((𝐴S𝐵S ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
2 shjval 31380 . . . 4 ((𝐵S𝐴S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐵𝐴))))
32ancoms 458 . . 3 ((𝐴S𝐵S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐵𝐴))))
4 uncom 4168 . . . . 5 (𝐵𝐴) = (𝐴𝐵)
54fveq2i 6910 . . . 4 (⊥‘(𝐵𝐴)) = (⊥‘(𝐴𝐵))
65fveq2i 6910 . . 3 (⊥‘(⊥‘(𝐵𝐴))) = (⊥‘(⊥‘(𝐴𝐵)))
73, 6eqtrdi 2791 . 2 ((𝐴S𝐵S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐴𝐵))))
81, 7eqtr4d 2778 1 ((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cun 3961  cfv 6563  (class class class)co 7431   S csh 30957  cort 30959   chj 30962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sh 31236  df-chj 31339
This theorem is referenced by:  shlej2  31390  shjcomi  31400  shub2  31412  chjcom  31535
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