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Theorem adjadd 29253
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 29048 . . 3 (𝑆 ∈ dom adj𝑆: ℋ⟶ ℋ)
2 dmadjop 29048 . . 3 (𝑇 ∈ dom adj𝑇: ℋ⟶ ℋ)
3 hoaddcl 28918 . . 3 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
41, 2, 3syl2an 495 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
5 dmadjrn 29055 . . . 4 (𝑆 ∈ dom adj → (adj𝑆) ∈ dom adj)
6 dmadjop 29048 . . . 4 ((adj𝑆) ∈ dom adj → (adj𝑆): ℋ⟶ ℋ)
75, 6syl 17 . . 3 (𝑆 ∈ dom adj → (adj𝑆): ℋ⟶ ℋ)
8 dmadjrn 29055 . . . 4 (𝑇 ∈ dom adj → (adj𝑇) ∈ dom adj)
9 dmadjop 29048 . . . 4 ((adj𝑇) ∈ dom adj → (adj𝑇): ℋ⟶ ℋ)
108, 9syl 17 . . 3 (𝑇 ∈ dom adj → (adj𝑇): ℋ⟶ ℋ)
11 hoaddcl 28918 . . 3 (((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ) → ((adj𝑆) +op (adj𝑇)): ℋ⟶ ℋ)
127, 10, 11syl2an 495 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ((adj𝑆) +op (adj𝑇)): ℋ⟶ ℋ)
13 adj2 29094 . . . . . . . 8 ((𝑆 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
14133expb 1113 . . . . . . 7 ((𝑆 ∈ dom adj ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
1514adantlr 753 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑆)‘𝑦)))
16 adj2 29094 . . . . . . . 8 ((𝑇 ∈ dom adj𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
17163expb 1113 . . . . . . 7 ((𝑇 ∈ dom adj ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
1817adantll 752 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih ((adj𝑇)‘𝑦)))
1915, 18oveq12d 6823 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)) = ((𝑥 ·ih ((adj𝑆)‘𝑦)) + (𝑥 ·ih ((adj𝑇)‘𝑦))))
201ffvelrnda 6514 . . . . . . 7 ((𝑆 ∈ dom adj𝑥 ∈ ℋ) → (𝑆𝑥) ∈ ℋ)
2120ad2ant2r 800 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑆𝑥) ∈ ℋ)
222ffvelrnda 6514 . . . . . . 7 ((𝑇 ∈ dom adj𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
2322ad2ant2lr 801 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇𝑥) ∈ ℋ)
24 simprr 813 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
25 ax-his2 28241 . . . . . 6 (((𝑆𝑥) ∈ ℋ ∧ (𝑇𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦) = (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)))
2621, 23, 24, 25syl3anc 1473 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦) = (((𝑆𝑥) ·ih 𝑦) + ((𝑇𝑥) ·ih 𝑦)))
27 simprl 811 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
28 adjcl 29092 . . . . . . 7 ((𝑆 ∈ dom adj𝑦 ∈ ℋ) → ((adj𝑆)‘𝑦) ∈ ℋ)
2928ad2ant2rl 802 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adj𝑆)‘𝑦) ∈ ℋ)
30 adjcl 29092 . . . . . . 7 ((𝑇 ∈ dom adj𝑦 ∈ ℋ) → ((adj𝑇)‘𝑦) ∈ ℋ)
3130ad2ant2l 799 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adj𝑇)‘𝑦) ∈ ℋ)
32 his7 28248 . . . . . 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 1473 . . . . 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 2797 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))) = (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦))
357, 10anim12i 591 . . . . . . 7 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ))
36 hosval 28900 . . . . . . . 8 (((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
37363expa 1111 . . . . . . 7 ((((adj𝑆): ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ) ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
3835, 37sylan 489 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ 𝑦 ∈ ℋ) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
3938adantrl 754 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((adj𝑆) +op (adj𝑇))‘𝑦) = (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦)))
4039oveq2d 6821 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)) = (𝑥 ·ih (((adj𝑆)‘𝑦) + ((adj𝑇)‘𝑦))))
411, 2anim12i 591 . . . . . . 7 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ))
42 hosval 28900 . . . . . . . 8 ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
43423expa 1111 . . . . . . 7 (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4441, 43sylan 489 . . . . . 6 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4544adantrr 755 . . . . 5 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆𝑥) + (𝑇𝑥)))
4645oveq1d 6820 . . . 4 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (((𝑆𝑥) + (𝑇𝑥)) ·ih 𝑦))
4734, 40, 463eqtr4rd 2797 . . 3 (((𝑆 ∈ dom adj𝑇 ∈ dom adj) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)))
4847ralrimivva 3101 . 2 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adj𝑆) +op (adj𝑇))‘𝑦)))
49 adjeq 29095 . 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 1473 1 ((𝑆 ∈ dom adj𝑇 ∈ dom adj) → (adj‘(𝑆 +op 𝑇)) = ((adj𝑆) +op (adj𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  wral 3042  dom cdm 5258  wf 6037  cfv 6041  (class class class)co 6805   + caddc 10123  chil 28077   + cva 28078   ·ih csp 28080   +op chos 28096  adjcado 28113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-hilex 28157  ax-hfvadd 28158  ax-hvcom 28159  ax-hvass 28160  ax-hv0cl 28161  ax-hvaddid 28162  ax-hfvmul 28163  ax-hvmulid 28164  ax-hvdistr2 28167  ax-hvmul0 28168  ax-hfi 28237  ax-his1 28240  ax-his2 28241  ax-his3 28242  ax-his4 28243
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-po 5179  df-so 5180  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-2 11263  df-cj 14030  df-re 14031  df-im 14032  df-hvsub 28129  df-hosum 28890  df-adjh 29009
This theorem is referenced by: (None)
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