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Theorem kbass3 31876
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 31707 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) ∈ β„‚)
21adantr 480 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((braβ€˜π΄)β€˜π΅) ∈ β„‚)
3 brafn 31705 . . . 4 (𝐢 ∈ β„‹ β†’ (braβ€˜πΆ): β„‹βŸΆβ„‚)
43ad2antrl 725 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ (braβ€˜πΆ): β„‹βŸΆβ„‚)
5 simprr 770 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ 𝐷 ∈ β„‹)
6 hfmval 31502 . . 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 1368 . 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 2732 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 395   = wceq 1533   ∈ wcel 2098  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  β„‚cc 11107   Β· cmul 11114   β„‹chba 30677   Β·fn chft 30700  bracbr 30714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-hilex 30757  ax-hfi 30837
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-hfmul 31492  df-bra 31608
This theorem is referenced by: (None)
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