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Theorem kbass3 31102
Description: Dirac bra-ket associative law ⟨𝐴 ∣ 𝐡⟩⟨𝐢 ∣ 𝐷⟩ = (⟨𝐴 ∣ 𝐡⟩⟨𝐢 ∣ ) ∣ 𝐷⟩. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
kbass3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π·)) = ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ))β€˜π·))

Proof of Theorem kbass3
StepHypRef Expression
1 bracl 30933 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) ∈ β„‚)
21adantr 482 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((braβ€˜π΄)β€˜π΅) ∈ β„‚)
3 brafn 30931 . . . 4 (𝐢 ∈ β„‹ β†’ (braβ€˜πΆ): β„‹βŸΆβ„‚)
43ad2antrl 727 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ (braβ€˜πΆ): β„‹βŸΆβ„‚)
5 simprr 772 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ 𝐷 ∈ β„‹)
6 hfmval 30728 . . 3 ((((braβ€˜π΄)β€˜π΅) ∈ β„‚ ∧ (braβ€˜πΆ): β„‹βŸΆβ„‚ ∧ 𝐷 ∈ β„‹) β†’ ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ))β€˜π·) = (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π·)))
72, 4, 5, 6syl3anc 1372 . 2 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ))β€˜π·) = (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π·)))
87eqcomd 2739 1 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π·)) = ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ))β€˜π·))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054   Β· cmul 11061   β„‹chba 29903   Β·fn chft 29926  bracbr 29940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-hilex 29983  ax-hfi 30063
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-hfmul 30718  df-bra 30834
This theorem is referenced by: (None)
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