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Theorem adjadd 31323
Description: The adjoint of the sum of two operators. Theorem 3.11(iii) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adjadd ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (adj‘(𝑆 +op 𝑇)) = ((adj𝑆) +op (adj𝑇)))

Proof of Theorem adjadd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmadjop 31118 . . 3 (𝑆 ∈ dom adj𝑆: ℋ⟶ ℋ)
2 dmadjop 31118 . . 3 (𝑇 ∈ dom adj𝑇: ℋ⟶ ℋ)
3 hoaddcl 30988 . . 3 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
41, 2, 3syl2an 597 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
5 dmadjrn 31125 . . . 4 (𝑆 ∈ dom adj → (adj𝑆) ∈ dom adj)
6 dmadjop 31118 . . . 4 ((adj𝑆) ∈ dom adj → (adj𝑆): ℋ⟶ ℋ)
75, 6syl 17 . . 3 (𝑆 ∈ dom adj → (adj𝑆): ℋ⟶ ℋ)
8 dmadjrn 31125 . . . 4 (𝑇 ∈ dom adj → (adj𝑇) ∈ dom adj)
9 dmadjop 31118 . . . 4 ((adj𝑇) ∈ dom adj → (adj𝑇): ℋ⟶ ℋ)
108, 9syl 17 . . 3 (𝑇 ∈ dom adj → (adj𝑇): ℋ⟶ ℋ)
11 hoaddcl 30988 . . 3 (((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ) → ((adj𝑆) +op (adj𝑇)): ℋ⟶ ℋ)
127, 10, 11syl2an 597 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ((adj𝑆) +op (adj𝑇)): ℋ⟶ ℋ)
13 adj2 31164 . . . . . . . 8 ((𝑆 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
14133expb 1121 . . . . . . 7 ((𝑆 ∈ dom adj ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
1514adantlr 714 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
16 adj2 31164 . . . . . . . 8 ((𝑇 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
17163expb 1121 . . . . . . 7 ((𝑇 ∈ dom adj ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
1817adantll 713 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
1915, 18oveq12d 7421 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)) = ((𝑥 ·ih ((adj𝑆)‘𝑦)) + (𝑥 ·ih ((adj𝑇)‘𝑦))))
201ffvelcdmda 7081 . . . . . . 7 ((𝑆 ∈ dom adj𝑥 ∈ ℋ) → (𝑆𝑥) ∈ ℋ)
2120ad2ant2r 746 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑆𝑥) ∈ ℋ)
222ffvelcdmda 7081 . . . . . . 7 ((𝑇 ∈ dom adj𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
2322ad2ant2lr 747 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇𝑥) ∈ ℋ)
24 simprr 772 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
25 ax-his2 30313 . . . . . 6 (((𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦) = (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)))
2621, 23, 24, 25syl3anc 1372 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦) = (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)))
27 simprl 770 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
28 adjcl 31162 . . . . . . 7 ((𝑆 ∈ dom adj𝑦 ∈ ℋ) → ((adj𝑆)‘𝑦) ∈ ℋ)
2928ad2ant2rl 748 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adj𝑆)‘𝑦) ∈ ℋ)
30 adjcl 31162 . . . . . . 7 ((𝑇 ∈ dom adj𝑦 ∈ ℋ) → ((adj𝑇)‘𝑦) ∈ ℋ)
3130ad2ant2l 745 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adj𝑇)‘𝑦) ∈ ℋ)
32 his7 30320 . . . . . 6 ((𝑥 ∈ ℋ ∧ ((adj𝑆)‘𝑦) ∈ ℋ ∧ ((adj𝑇)‘𝑦) ∈ ℋ) → (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))) = ((𝑥 ·ih ((adj𝑆)‘𝑦)) + (𝑥 ·ih ((adj𝑇)‘𝑦))))
3327, 29, 31, 32syl3anc 1372 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))) = ((𝑥 ·ih ((adj𝑆)‘𝑦)) + (𝑥 ·ih ((adj𝑇)‘𝑦))))
3419, 26, 333eqtr4rd 2784 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))) = (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦))
357, 10anim12i 614 . . . . . . 7 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ))
36 hosval 30970 . . . . . . . 8 (((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
37363expa 1119 . . . . . . 7 ((((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ) ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
3835, 37sylan 581 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
3938adantrl 715 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
4039oveq2d 7419 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)) = (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))))
411, 2anim12i 614 . . . . . . 7 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ))
42 hosval 30970 . . . . . . . 8 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
43423expa 1119 . . . . . . 7 (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4441, 43sylan 581 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4544adantrr 716 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4645oveq1d 7418 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦))
4734, 40, 463eqtr4rd 2784 . . 3 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)))
4847ralrimivva 3201 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)))
49 adjeq 31165 . 2 (((𝑆 +op 𝑇): ℋ⟶ ℋ ∧ ((adj𝑆) +op (adj𝑇)): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦))) → (adj‘(𝑆 +op 𝑇)) = ((adj𝑆) +op (adj𝑇)))
504, 12, 48, 49syl3anc 1372 1 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (adj‘(𝑆 +op 𝑇)) = ((adj𝑆) +op (adj𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  dom cdm 5674  wf 6535  cfv 6539  (class class class)co 7403   + caddc 11108  chba 30149   + cva 30150   ·ih csp 30152   +op chos 30168  adjcado 30185
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 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-hilex 30229  ax-hfvadd 30230  ax-hvcom 30231  ax-hvass 30232  ax-hv0cl 30233  ax-hvaddid 30234  ax-hfvmul 30235  ax-hvmulid 30236  ax-hvdistr2 30239  ax-hvmul0 30240  ax-hfi 30309  ax-his1 30312  ax-his2 30313  ax-his3 30314  ax-his4 30315
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-po 5586  df-so 5587  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7359  df-ov 7406  df-oprab 7407  df-mpo 7408  df-er 8698  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11441  df-neg 11442  df-div 11867  df-2 12270  df-cj 15041  df-re 15042  df-im 15043  df-hvsub 30201  df-hosum 30960  df-adjh 31079
This theorem is referenced by: (None)
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