Theorem lnophsi 32072

Description: The sum of two linear operators is linear. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lnopco.1	⊢ 𝑆 ∈ LinOp
lnopco.2	⊢ 𝑇 ∈ LinOp

Assertion

Ref	Expression
lnophsi	⊢ (𝑆 +op 𝑇) ∈ LinOp

Proof of Theorem lnophsi

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

Step	Hyp	Ref	Expression
1		lnopco.1	. . . 4 ⊢ 𝑆 ∈ LinOp
2	1	lnopfi 32040	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 32040	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hoaddcli 31839	. 2 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
6		hvmulcl 31084	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
7	1	lnopaddi 32042	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑆‘𝑧)))
8	3	lnopaddi 32042	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑇‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘𝑧)))
9	7, 8	oveq12d 7385	. . . . . . 7 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) +ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑆‘𝑧)) +ℎ ((𝑇‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘𝑧))))
10	6, 9	sylan 581	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) +ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑆‘𝑧)) +ℎ ((𝑇‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘𝑧))))
11	2	ffvelcdmi 7035	. . . . . . . . 9 ⊢ ((𝑥 ·ℎ 𝑦) ∈ ℋ → (𝑆‘(𝑥 ·ℎ 𝑦)) ∈ ℋ)
12	6, 11	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·ℎ 𝑦)) ∈ ℋ)
13	2	ffvelcdmi 7035	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑆‘𝑧) ∈ ℋ)
14	12, 13	anim12i 614	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆‘(𝑥 ·ℎ 𝑦)) ∈ ℋ ∧ (𝑆‘𝑧) ∈ ℋ))
15	4	ffvelcdmi 7035	. . . . . . . . 9 ⊢ ((𝑥 ·ℎ 𝑦) ∈ ℋ → (𝑇‘(𝑥 ·ℎ 𝑦)) ∈ ℋ)
16	6, 15	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·ℎ 𝑦)) ∈ ℋ)
17	4	ffvelcdmi 7035	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
18	16, 17	anim12i 614	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑇‘(𝑥 ·ℎ 𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ))
19		hvadd4 31107	. . . . . . 7 ⊢ ((((𝑆‘(𝑥 ·ℎ 𝑦)) ∈ ℋ ∧ (𝑆‘𝑧) ∈ ℋ) ∧ ((𝑇‘(𝑥 ·ℎ 𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ)) → (((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑆‘𝑧)) +ℎ ((𝑇‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘𝑧))) = (((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘(𝑥 ·ℎ 𝑦))) +ℎ ((𝑆‘𝑧) +ℎ (𝑇‘𝑧))))
20	14, 18, 19	syl2anc 585	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑆‘𝑧)) +ℎ ((𝑇‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘𝑧))) = (((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘(𝑥 ·ℎ 𝑦))) +ℎ ((𝑆‘𝑧) +ℎ (𝑇‘𝑧))))
21	10, 20	eqtrd 2771	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) +ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))) = (((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘(𝑥 ·ℎ 𝑦))) +ℎ ((𝑆‘𝑧) +ℎ (𝑇‘𝑧))))
22		hvaddcl 31083	. . . . . . 7 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
23	6, 22	sylan 581	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
24		hosval 31811	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) → ((𝑆 +op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑆‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) +ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))))
25	2, 4, 24	mp3an12 1454	. . . . . 6 ⊢ (((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ → ((𝑆 +op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑆‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) +ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))))
26	23, 25	syl 17	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆 +op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑆‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) +ℎ (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧))))
27	2	ffvelcdmi 7035	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑆‘𝑦) ∈ ℋ)
28	4	ffvelcdmi 7035	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
29	27, 28	jca 511	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
30		ax-hvdistr1 31079	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))) = ((𝑥 ·ℎ (𝑆‘𝑦)) +ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
31	30	3expb 1121	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ)) → (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))) = ((𝑥 ·ℎ (𝑆‘𝑦)) +ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
32	29, 31	sylan2 594	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))) = ((𝑥 ·ℎ (𝑆‘𝑦)) +ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
33		hosval 31811	. . . . . . . . . 10 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑦) = ((𝑆‘𝑦) +ℎ (𝑇‘𝑦)))
34	2, 4, 33	mp3an12 1454	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑦) = ((𝑆‘𝑦) +ℎ (𝑇‘𝑦)))
35	34	oveq2d 7383	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))))
36	35	adantl 481	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))))
37	1	lnopmuli 32043	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·ℎ 𝑦)) = (𝑥 ·ℎ (𝑆‘𝑦)))
38	3	lnopmuli 32043	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·ℎ 𝑦)) = (𝑥 ·ℎ (𝑇‘𝑦)))
39	37, 38	oveq12d 7385	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘(𝑥 ·ℎ 𝑦))) = ((𝑥 ·ℎ (𝑆‘𝑦)) +ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
40	32, 36, 39	3eqtr4d 2781	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = ((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘(𝑥 ·ℎ 𝑦))))
41		hosval 31811	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑧) = ((𝑆‘𝑧) +ℎ (𝑇‘𝑧)))
42	2, 4, 41	mp3an12 1454	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑧) = ((𝑆‘𝑧) +ℎ (𝑇‘𝑧)))
43	40, 42	oveqan12d 7386	. . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) +ℎ ((𝑆 +op 𝑇)‘𝑧)) = (((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘(𝑥 ·ℎ 𝑦))) +ℎ ((𝑆‘𝑧) +ℎ (𝑇‘𝑧))))
44	21, 26, 43	3eqtr4d 2781	. . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆 +op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) +ℎ ((𝑆 +op 𝑇)‘𝑧)))
45	44	ralrimiva 3129	. . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ∀𝑧 ∈ ℋ ((𝑆 +op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) +ℎ ((𝑆 +op 𝑇)‘𝑧)))
46	45	rgen2 3177	. 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆 +op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) +ℎ ((𝑆 +op 𝑇)‘𝑧))
47		ellnop 31929	. 2 ⊢ ((𝑆 +op 𝑇) ∈ LinOp ↔ ((𝑆 +op 𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆 +op 𝑇)‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) +ℎ ((𝑆 +op 𝑇)‘𝑧))))
48	5, 46, 47	mpbir2an 712	1 ⊢ (𝑆 +op 𝑇) ∈ LinOp

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ℂcc 11036   ℋchba 30990   +_ℎ cva 30991   ·_ℎ csm 30992   +_op chos 31009  LinOpclo 31018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-hilex 31070  ax-hfvadd 31071  ax-hvcom 31072  ax-hvass 31073  ax-hv0cl 31074  ax-hvaddid 31075  ax-hfvmul 31076  ax-hvmulid 31077  ax-hvdistr1 31079  ax-hvdistr2 31080  ax-hvmul0 31081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-ltxr 11184  df-sub 11379  df-neg 11380  df-hvsub 31042  df-hosum 31801  df-lnop 31912

This theorem is referenced by:  lnophdi  32073  bdophsi  32167