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 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 31118	. . 3 ⊢ (𝑆 ∈ dom adjℎ → 𝑆: ℋ⟶ ℋ)
2		dmadjop 31118	. . 3 ⊢ (𝑇 ∈ dom adjℎ → 𝑇: ℋ⟶ ℋ)
3		hoaddcl 30988	. . 3 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
4	1, 2, 3	syl2an 597	. 2 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (𝑆 +op 𝑇): ℋ⟶ ℋ)
5		dmadjrn 31125	. . . 4 ⊢ (𝑆 ∈ dom adjℎ → (adjℎ‘𝑆) ∈ dom adjℎ)
6		dmadjop 31118	. . . 4 ⊢ ((adjℎ‘𝑆) ∈ dom adjℎ → (adjℎ‘𝑆): ℋ⟶ ℋ)
7	5, 6	syl 17	. . 3 ⊢ (𝑆 ∈ dom adjℎ → (adjℎ‘𝑆): ℋ⟶ ℋ)
8		dmadjrn 31125	. . . 4 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ)
9		dmadjop 31118	. . . 4 ⊢ ((adjℎ‘𝑇) ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
10	8, 9	syl 17	. . 3 ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇): ℋ⟶ ℋ)
11		hoaddcl 30988	. . 3 ⊢ (((adjℎ‘𝑆): ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ) → ((adjℎ‘𝑆) +op (adjℎ‘𝑇)): ℋ⟶ ℋ)
12	7, 10, 11	syl2an 597	. 2 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → ((adjℎ‘𝑆) +op (adjℎ‘𝑇)): ℋ⟶ ℋ)
13		adj2 31164	. . . . . . . 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 714	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑆)‘𝑦)))
16		adj2 31164	. . . . . . . 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 713	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih ((adjℎ‘𝑇)‘𝑦)))
19	15, 18	oveq12d 7421	. . . . 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 746	. . . . . 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 747	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑥) ∈ ℋ)
24		simprr 772	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
25		ax-his2 30313	. . . . . 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 1372	. . . . 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 770	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
28		adjcl 31162	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → ((adjℎ‘𝑆)‘𝑦) ∈ ℋ)
29	28	ad2ant2rl 748	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adjℎ‘𝑆)‘𝑦) ∈ ℋ)
30		adjcl 31162	. . . . . . 7 ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝑦 ∈ ℋ) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
31	30	ad2ant2l 745	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((adjℎ‘𝑇)‘𝑦) ∈ ℋ)
32		his7 30320	. . . . . 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 1372	. . . . 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 2784	. . . 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 614	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → ((adjℎ‘𝑆): ℋ⟶ ℋ ∧ (adjℎ‘𝑇): ℋ⟶ ℋ))
36		hosval 30970	. . . . . . . 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 581	. . . . . 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 715	. . . . 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 7419	. . . 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 614	. . . . . . 7 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ))
42		hosval 30970	. . . . . . . 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 581	. . . . . 6 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
45	44	adantrr 716	. . . . 5 ⊢ (((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
46	45	oveq1d 7418	. . . 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 2784	. . 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 3201	. 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 31165	. 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 1372	1 ⊢ ((𝑆 ∈ dom adjℎ ∧ 𝑇 ∈ dom adjℎ) → (adjℎ‘(𝑆 +op 𝑇)) = ((adjℎ‘𝑆) +op (adjℎ‘𝑇)))

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   ℋchba 30149   +_ℎ cva 30150   ·_ih csp 30152   +_op chos 30168  adj_ℎcado 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)