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Theorem shjcom 29148
 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 29141 . 2 ((𝐴S𝐵S ) → (𝐴 𝐵) = (⊥‘(⊥‘(𝐴𝐵))))
2 shjval 29141 . . . 4 ((𝐵S𝐴S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐵𝐴))))
32ancoms 462 . . 3 ((𝐴S𝐵S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐵𝐴))))
4 uncom 4080 . . . . 5 (𝐵𝐴) = (𝐴𝐵)
54fveq2i 6648 . . . 4 (⊥‘(𝐵𝐴)) = (⊥‘(𝐴𝐵))
65fveq2i 6648 . . 3 (⊥‘(⊥‘(𝐵𝐴))) = (⊥‘(⊥‘(𝐴𝐵)))
73, 6eqtrdi 2849 . 2 ((𝐴S𝐵S ) → (𝐵 𝐴) = (⊥‘(⊥‘(𝐴𝐵))))
81, 7eqtr4d 2836 1 ((𝐴S𝐵S ) → (𝐴 𝐵) = (𝐵 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∪ cun 3879  ‘cfv 6324  (class class class)co 7135   Sℋ csh 28718  ⊥cort 28720   ∨ℋ chj 28723 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-hilex 28789 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-sh 28997  df-chj 29100 This theorem is referenced by:  shlej2  29151  shjcomi  29161  shub2  29173  chjcom  29296
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