Theorem adjadd 32112

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 31907	. . 3 ⊢ (𝑆 ∈ dom adjℎ → 𝑆: ℋ⟶ ℋ)
2		dmadjop 31907	. . 3 ⊢ (𝑇 ∈ dom adjℎ → 𝑇: ℋ⟶ ℋ)
3		hoaddcl 31777	. . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
5		dmadjrn 31914	. . . 4 ⊢ (𝑆 ∈ dom adjℎ → (adjℎ‘𝑆) ∈ dom adjℎ)
6		dmadjop 31907	. . . 4 ⊢ ((adjℎ‘𝑆) ∈ dom adjℎ → (adjℎ‘𝑆): ℋ⟶ ℋ)
7	5, 6	syl 17	. . 3 ⊢ (𝑆 ∈ dom adjℎ → (adjℎ‘𝑆): ℋ⟶ ℋ)
8		dmadjrn 31914	. . . 4 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ)
9		dmadjop 31907	. . . 4 ⊢ ((adjℎ‘𝑇) ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
10	8, 9	syl 17	. . 3 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
11		hoaddcl 31777	. . 3 ⊢ (((adjℎ‘𝑆): ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ) → ((adjℎ‘𝑆) +op (adjℎ‘𝑇)): ℋ⟶ ℋ)
12	7, 10, 11	syl2an 596	. 2 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → ((adjℎ‘𝑆) +op (adjℎ‘𝑇)): ℋ⟶ ℋ)
13		adj2 31953	. . . . . . . 8 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)))
14	13	3expb 1121	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)))
15	14	adantlr 715	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)))
16		adj2 31953	. . . . . . . 8 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
17	16	3expb 1121	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
18	17	adantll 714	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
19	15, 18	oveq12d 7449	. . . . 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 7104	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑆‘𝑥) ∈ ℋ)
21	20	ad2ant2r 747	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑆‘𝑥) ∈ ℋ)
22	2	ffvelcdmda 7104	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
23	22	ad2ant2lr 748	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑥) ∈ ℋ)
24		simprr 773	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
25		ax-his2 31102	. . . . . 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 1373	. . . . 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 771	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
28		adjcl 31951	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → ((adjℎ‘𝑆)‘𝑦) ∈ ℋ)
29	28	ad2ant2rl 749	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adjℎ‘𝑆)‘𝑦) ∈ ℋ)
30		adjcl 31951	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
31	30	ad2ant2l 746	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
32		his7 31109	. . . . . 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 1373	. . . . 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 2788	. . . 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 613	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → ((adjℎ‘𝑆): ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ))
36		hosval 31759	. . . . . . . 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 1119	. . . . . . 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 580	. . . . . 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 716	. . . . 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 7447	. . . 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 613	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ))
42		hosval 31759	. . . . . . . 8 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
43	42	3expa 1119	. . . . . . 7 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
44	41, 43	sylan 580	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
45	44	adantrr 717	. . . . 5 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
46	45	oveq1d 7446	. . . 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 2788	. . 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 3202	. 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 31954	. 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 1373	1 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝑆 +op 𝑇)) = ((adjℎ‘𝑆) +op (adjℎ‘𝑇)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  dom cdm 5685  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   + caddc 11158   ℋchba 30938   +_ℎ cva 30939   ·_ih csp 30941   +_op chos 30957  adj_ℎcado 30974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-hilex 31018  ax-hfvadd 31019  ax-hvcom 31020  ax-hvass 31021  ax-hv0cl 31022  ax-hvaddid 31023  ax-hfvmul 31024  ax-hvmulid 31025  ax-hvdistr2 31028  ax-hvmul0 31029  ax-hfi 31098  ax-his1 31101  ax-his2 31102  ax-his3 31103  ax-his4 31104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-2 12329  df-cj 15138  df-re 15139  df-im 15140  df-hvsub 30990  df-hosum 31749  df-adjh 31868

This theorem is referenced by: (None)