Theorem adjadd 32426

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 32221	. . 3 ⊢ (𝑆 ∈ dom adjℎ → 𝑆: ℋ⟶ ℋ)
2		dmadjop 32221	. . 3 ⊢ (𝑇 ∈ dom adjℎ → 𝑇: ℋ⟶ ℋ)
3		hoaddcl 32091	. . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
4	1, 2, 3	syl2an 607	. 2 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
5		dmadjrn 32228	. . . 4 ⊢ (𝑆 ∈ dom adjℎ → (adjℎ‘𝑆) ∈ dom adjℎ)
6		dmadjop 32221	. . . 4 ⊢ ((adjℎ‘𝑆) ∈ dom adjℎ → (adjℎ‘𝑆): ℋ⟶ ℋ)
7	5, 6	syl 18	. . 3 ⊢ (𝑆 ∈ dom adjℎ → (adjℎ‘𝑆): ℋ⟶ ℋ)
8		dmadjrn 32228	. . . 4 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ)
9		dmadjop 32221	. . . 4 ⊢ ((adjℎ‘𝑇) ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
10	8, 9	syl 18	. . 3 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
11		hoaddcl 32091	. . 3 ⊢ (((adjℎ‘𝑆): ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ) → ((adjℎ‘𝑆) +op (adjℎ‘𝑇)): ℋ⟶ ℋ)
12	7, 10, 11	syl2an 607	. 2 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → ((adjℎ‘𝑆) +op (adjℎ‘𝑇)): ℋ⟶ ℋ)
13		adj2 32267	. . . . . . . 8 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)))
14	13	3expb 1138	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)))
15	14	adantlr 727	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)))
16		adj2 32267	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
17	16	3expb 1138	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
18	17	adantll 726	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
19	15, 18	oveq12d 7430	. . . . 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 7081	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑆‘𝑥) ∈ ℋ)
21	20	ad2ant2r 759	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑆‘𝑥) ∈ ℋ)
22	2	ffvelcdmda 7081	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
23	22	ad2ant2lr 760	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑥) ∈ ℋ)
24		simprr 784	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
25		ax-his2 31416	. . . . . 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 1398	. . . . 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 782	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
28		adjcl 32265	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → ((adjℎ‘𝑆)‘𝑦) ∈ ℋ)
29	28	ad2ant2rl 761	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adjℎ‘𝑆)‘𝑦) ∈ ℋ)
30		adjcl 32265	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
31	30	ad2ant2l 758	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
32		his7 31423	. . . . . 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 1398	. . . . 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 2809	. . . 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 624	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → ((adjℎ‘𝑆): ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ))
36		hosval 32073	. . . . . . . 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 1136	. . . . . . 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 591	. . . . . 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 728	. . . . 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 7428	. . . 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 624	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ))
42		hosval 32073	. . . . . . . 8 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
43	42	3expa 1136	. . . . . . 7 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
44	41, 43	sylan 591	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
45	44	adantrr 729	. . . . 5 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
46	45	oveq1d 7427	. . . 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 2809	. . 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 3208	. 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 32268	. 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 1398	1 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝑆 +op 𝑇)) = ((adjℎ‘𝑆) +op (adjℎ‘𝑇)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  dom cdm 5663  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   + caddc 11104   ℋchba 31252   +_ℎ cva 31253   ·_ih csp 31255   +_op chos 31271  adj_ℎcado 31288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-hilex 31332  ax-hfvadd 31333  ax-hvcom 31334  ax-hvass 31335  ax-hv0cl 31336  ax-hvaddid 31337  ax-hfvmul 31338  ax-hvmulid 31339  ax-hvdistr2 31342  ax-hvmul0 31343  ax-hfi 31412  ax-his1 31415  ax-his2 31416  ax-his3 31417  ax-his4 31418

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-cj 15152  df-re 15153  df-im 15154  df-hvsub 31304  df-hosum 32063  df-adjh 32182

This theorem is referenced by: (None)