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Theorem hoadddi 29565
Description: Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hoadddi ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)))

Proof of Theorem hoadddi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpl1 1187 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℂ)
2 ffvelrn 6825 . . . . . . 7 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
323ad2antl2 1182 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
4 ffvelrn 6825 . . . . . . 7 ((𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑈𝑥) ∈ ℋ)
543ad2antl3 1183 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑈𝑥) ∈ ℋ)
6 ax-hvdistr1 28770 . . . . . 6 ((𝐴 ∈ ℂ ∧ (𝑇𝑥) ∈ ℋ ∧ (𝑈𝑥) ∈ ℋ) → (𝐴 · ((𝑇𝑥) + (𝑈𝑥))) = ((𝐴 · (𝑇𝑥)) + (𝐴 · (𝑈𝑥))))
71, 3, 5, 6syl3anc 1367 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 · ((𝑇𝑥) + (𝑈𝑥))) = ((𝐴 · (𝑇𝑥)) + (𝐴 · (𝑈𝑥))))
8 hosval 29502 . . . . . . . 8 ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑈)‘𝑥) = ((𝑇𝑥) + (𝑈𝑥)))
98oveq2d 7149 . . . . . . 7 ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝐴 · ((𝑇 +op 𝑈)‘𝑥)) = (𝐴 · ((𝑇𝑥) + (𝑈𝑥))))
1093expa 1114 . . . . . 6 (((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 · ((𝑇 +op 𝑈)‘𝑥)) = (𝐴 · ((𝑇𝑥) + (𝑈𝑥))))
11103adantl1 1162 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 · ((𝑇 +op 𝑈)‘𝑥)) = (𝐴 · ((𝑇𝑥) + (𝑈𝑥))))
12 homval 29503 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 · (𝑇𝑥)))
13123expa 1114 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 · (𝑇𝑥)))
14133adantl3 1164 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑥) = (𝐴 · (𝑇𝑥)))
15 homval 29503 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑈)‘𝑥) = (𝐴 · (𝑈𝑥)))
16153expa 1114 . . . . . . 7 (((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑈)‘𝑥) = (𝐴 · (𝑈𝑥)))
17163adantl2 1163 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op 𝑈)‘𝑥) = (𝐴 · (𝑈𝑥)))
1814, 17oveq12d 7151 . . . . 5 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇)‘𝑥) + ((𝐴 ·op 𝑈)‘𝑥)) = ((𝐴 · (𝑇𝑥)) + (𝐴 · (𝑈𝑥))))
197, 11, 183eqtr4d 2865 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 · ((𝑇 +op 𝑈)‘𝑥)) = (((𝐴 ·op 𝑇)‘𝑥) + ((𝐴 ·op 𝑈)‘𝑥)))
20 hoaddcl 29520 . . . . . . 7 ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 +op 𝑈): ℋ⟶ ℋ)
2120anim2i 618 . . . . . 6 ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → (𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ))
22213impb 1111 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ))
23 homval 29503 . . . . . 6 ((𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (𝐴 · ((𝑇 +op 𝑈)‘𝑥)))
24233expa 1114 . . . . 5 (((𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (𝐴 · ((𝑇 +op 𝑈)‘𝑥)))
2522, 24sylan 582 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (𝐴 · ((𝑇 +op 𝑈)‘𝑥)))
26 homulcl 29521 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
27 homulcl 29521 . . . . . . 7 ((𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op 𝑈): ℋ⟶ ℋ)
2826, 27anim12i 614 . . . . . 6 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ))
29283impdi 1346 . . . . 5 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ))
30 hosval 29502 . . . . . 6 (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇)‘𝑥) + ((𝐴 ·op 𝑈)‘𝑥)))
31303expa 1114 . . . . 5 ((((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇)‘𝑥) + ((𝐴 ·op 𝑈)‘𝑥)))
3229, 31sylan 582 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇)‘𝑥) + ((𝐴 ·op 𝑈)‘𝑥)))
3319, 25, 323eqtr4d 2865 . . 3 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥))
3433ralrimiva 3169 . 2 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ∀𝑥 ∈ ℋ ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥))
35 homulcl 29521 . . . . 5 ((𝐴 ∈ ℂ ∧ (𝑇 +op 𝑈): ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)): ℋ⟶ ℋ)
3620, 35sylan2 594 . . . 4 ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ)) → (𝐴 ·op (𝑇 +op 𝑈)): ℋ⟶ ℋ)
37363impb 1111 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)): ℋ⟶ ℋ)
38 hoaddcl 29520 . . . . 5 (((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ (𝐴 ·op 𝑈): ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)): ℋ⟶ ℋ)
3926, 27, 38syl2an 597 . . . 4 (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝐴 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ)) → ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)): ℋ⟶ ℋ)
40393impdi 1346 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)): ℋ⟶ ℋ)
41 hoeq 29522 . . 3 (((𝐴 ·op (𝑇 +op 𝑈)): ℋ⟶ ℋ ∧ ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)): ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥) ↔ (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))))
4237, 40, 41syl2anc 586 . 2 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ((𝐴 ·op (𝑇 +op 𝑈))‘𝑥) = (((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))‘𝑥) ↔ (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈))))
4334, 42mpbid 234 1 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝐴 ·op (𝑇 +op 𝑈)) = ((𝐴 ·op 𝑇) +op (𝐴 ·op 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wral 3125  wf 6327  cfv 6331  (class class class)co 7133  cc 10513  chba 28681   + cva 28682   · csm 28683   +op chos 28700   ·op chot 28701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-hilex 28761  ax-hfvadd 28762  ax-hfvmul 28767  ax-hvdistr1 28770
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-map 8386  df-hosum 29492  df-homul 29493
This theorem is referenced by:  hosubdi  29570  honegdi  29571  ho2times  29581  opsqrlem6  29907
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