Theorem adjadd 32189

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 adjℎ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝑆 +op 𝑇)) = ((adjℎ‘𝑆) +op (adjℎ‘𝑇)))

Proof of Theorem adjadd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmadjop 31984	. . 3 ⊢ (𝑆 ∈ dom adjℎ → 𝑆: ℋ⟶ ℋ)
2		dmadjop 31984	. . 3 ⊢ (𝑇 ∈ dom adjℎ → 𝑇: ℋ⟶ ℋ)
3		hoaddcl 31854	. . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
4	1, 2, 3	syl2an 602	. 2 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
5		dmadjrn 31991	. . . 4 ⊢ (𝑆 ∈ dom adjℎ → (adjℎ‘𝑆) ∈ dom adjℎ)
6		dmadjop 31984	. . . 4 ⊢ ((adjℎ‘𝑆) ∈ dom adjℎ → (adjℎ‘𝑆): ℋ⟶ ℋ)
7	5, 6	syl 17	. . 3 ⊢ (𝑆 ∈ dom adjℎ → (adjℎ‘𝑆): ℋ⟶ ℋ)
8		dmadjrn 31991	. . . 4 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ)
9		dmadjop 31984	. . . 4 ⊢ ((adjℎ‘𝑇) ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
10	8, 9	syl 17	. . 3 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
11		hoaddcl 31854	. . 3 ⊢ (((adjℎ‘𝑆): ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ) → ((adjℎ‘𝑆) +op (adjℎ‘𝑇)): ℋ⟶ ℋ)
12	7, 10, 11	syl2an 602	. 2 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → ((adjℎ‘𝑆) +op (adjℎ‘𝑇)): ℋ⟶ ℋ)
13		adj2 32030	. . . . . . . 8 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)))
14	13	3expb 1126	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)))
15	14	adantlr 721	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)))
16		adj2 32030	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
17	16	3expb 1126	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
18	17	adantll 720	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
19	15, 18	oveq12d 7381	. . . . 5 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆‘𝑥) ·ih 𝑦) + ((𝑇‘𝑥) ·ih 𝑦)) = ((𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)) + (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦))))
20	1	ffvelcdmda 7032	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑆‘𝑥) ∈ ℋ)
21	20	ad2ant2r 753	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑆‘𝑥) ∈ ℋ)
22	2	ffvelcdmda 7032	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
23	22	ad2ant2lr 754	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑥) ∈ ℋ)
24		simprr 778	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
25		ax-his2 31179	. . . . . 6 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) ·ih 𝑦) = (((𝑆‘𝑥) ·ih 𝑦) + ((𝑇‘𝑥) ·ih 𝑦)))
26	21, 23, 24, 25	syl3anc 1379	. . . . 5 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) ·ih 𝑦) = (((𝑆‘𝑥) ·ih 𝑦) + ((𝑇‘𝑥) ·ih 𝑦)))
27		simprl 776	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
28		adjcl 32028	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → ((adjℎ‘𝑆)‘𝑦) ∈ ℋ)
29	28	ad2ant2rl 755	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adjℎ‘𝑆)‘𝑦) ∈ ℋ)
30		adjcl 32028	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
31	30	ad2ant2l 752	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
32		his7 31186	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ ((adjℎ‘𝑆)‘𝑦) ∈ ℋ ∧ ((adjℎ‘𝑇)‘𝑦) ∈ ℋ) → (𝑥 ·ih (((adjℎ‘𝑆)‘𝑦) +ℎ ((adjℎ‘𝑇)‘𝑦))) = ((𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)) + (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦))))
33	27, 29, 31, 32	syl3anc 1379	. . . . 5 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adjℎ‘𝑆)‘𝑦) +ℎ ((adjℎ‘𝑇)‘𝑦))) = ((𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)) + (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦))))
34	19, 26, 33	3eqtr4rd 2786	. . . 4 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adjℎ‘𝑆)‘𝑦) +ℎ ((adjℎ‘𝑇)‘𝑦))) = (((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) ·ih 𝑦))
35	7, 10	anim12i 619	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → ((adjℎ‘𝑆): ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ))
36		hosval 31836	. . . . . . . 8 ⊢ (((adjℎ‘𝑆): ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (((adjℎ‘𝑆) +op (adjℎ‘𝑇))‘𝑦) = (((adjℎ‘𝑆)‘𝑦) +ℎ ((adjℎ‘𝑇)‘𝑦)))
37	36	3expa 1124	. . . . . . 7 ⊢ ((((adjℎ‘𝑆): ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ) ∧ 𝑦 ∈ ℋ) → (((adjℎ‘𝑆) +op (adjℎ‘𝑇))‘𝑦) = (((adjℎ‘𝑆)‘𝑦) +ℎ ((adjℎ‘𝑇)‘𝑦)))
38	35, 37	sylan 586	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ 𝑦 ∈ ℋ) → (((adjℎ‘𝑆) +op (adjℎ‘𝑇))‘𝑦) = (((adjℎ‘𝑆)‘𝑦) +ℎ ((adjℎ‘𝑇)‘𝑦)))
39	38	adantrl 722	. . . . 5 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((adjℎ‘𝑆) +op (adjℎ‘𝑇))‘𝑦) = (((adjℎ‘𝑆)‘𝑦) +ℎ ((adjℎ‘𝑇)‘𝑦)))
40	39	oveq2d 7379	. . . 4 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (((adjℎ‘𝑆) +op (adjℎ‘𝑇))‘𝑦)) = (𝑥 ·ih (((adjℎ‘𝑆)‘𝑦) +ℎ ((adjℎ‘𝑇)‘𝑦))))
41	1, 2	anim12i 619	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ))
42		hosval 31836	. . . . . . . 8 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
43	42	3expa 1124	. . . . . . 7 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
44	41, 43	sylan 586	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
45	44	adantrr 723	. . . . 5 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
46	45	oveq1d 7378	. . . 4 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) ·ih 𝑦))
47	34, 40, 46	3eqtr4rd 2786	. . 3 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adjℎ‘𝑆) +op (adjℎ‘𝑇))‘𝑦)))
48	47	ralrimivva 3183	. 2 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adjℎ‘𝑆) +op (adjℎ‘𝑇))‘𝑦)))
49		adjeq 32031	. 2 ⊢ (((𝑆 +op 𝑇): ℋ⟶ ℋ ∧ ((adjℎ‘𝑆) +op (adjℎ‘𝑇)): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 +op 𝑇)‘𝑥) ·ih 𝑦) = (𝑥 ·ih (((adjℎ‘𝑆) +op (adjℎ‘𝑇))‘𝑦))) → (adjℎ‘(𝑆 +op 𝑇)) = ((adjℎ‘𝑆) +op (adjℎ‘𝑇)))
50	4, 12, 48, 49	syl3anc 1379	1 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝑆 +op 𝑇)) = ((adjℎ‘𝑆) +op (adjℎ‘𝑇)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  dom cdm 5625  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   + caddc 11039   ℋchba 31015   +_ℎ cva 31016   ·_ih csp 31018   +_op chos 31034  adj_ℎcado 31051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-hilex 31095  ax-hfvadd 31096  ax-hvcom 31097  ax-hvass 31098  ax-hv0cl 31099  ax-hvaddid 31100  ax-hfvmul 31101  ax-hvmulid 31102  ax-hvdistr2 31105  ax-hvmul0 31106  ax-hfi 31175  ax-his1 31178  ax-his2 31179  ax-his3 31180  ax-his4 31181

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-cj 15059  df-re 15060  df-im 15061  df-hvsub 31067  df-hosum 31826  df-adjh 31945

This theorem is referenced by: (None)