Theorem lnophsi 32353

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 32321	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 32321	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hoaddcli 32120	. 2 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
6		hvmulcl 31365	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
7	1	lnopaddi 32323	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑆‘𝑧)))
8	3	lnopaddi 32323	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑇‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘𝑧)))
9	7, 8	oveq12d 7428	. . . . . . 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 591	. . . . . 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 7078	. . . . . . . . 9 ⊢ ((𝑥 ·ℎ 𝑦) ∈ ℋ → (𝑆‘(𝑥 ·ℎ 𝑦)) ∈ ℋ)
12	6, 11	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·ℎ 𝑦)) ∈ ℋ)
13	2	ffvelcdmi 7078	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑆‘𝑧) ∈ ℋ)
14	12, 13	anim12i 624	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆‘(𝑥 ·ℎ 𝑦)) ∈ ℋ ∧ (𝑆‘𝑧) ∈ ℋ))
15	4	ffvelcdmi 7078	. . . . . . . . 9 ⊢ ((𝑥 ·ℎ 𝑦) ∈ ℋ → (𝑇‘(𝑥 ·ℎ 𝑦)) ∈ ℋ)
16	6, 15	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·ℎ 𝑦)) ∈ ℋ)
17	4	ffvelcdmi 7078	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
18	16, 17	anim12i 624	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑇‘(𝑥 ·ℎ 𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ))
19		hvadd4 31388	. . . . . . 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 595	. . . . . 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 2798	. . . . 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 31364	. . . . . . 7 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
23	6, 22	sylan 591	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
24		hosval 32092	. . . . . . 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 1480	. . . . . 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 18	. . . . 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 7078	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑆‘𝑦) ∈ ℋ)
28	4	ffvelcdmi 7078	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
29	27, 28	jca 520	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
30		ax-hvdistr1 31360	. . . . . . . . 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 1138	. . . . . . . 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 604	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))) = ((𝑥 ·ℎ (𝑆‘𝑦)) +ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
33		hosval 32092	. . . . . . . . . 10 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑦) = ((𝑆‘𝑦) +ℎ (𝑇‘𝑦)))
34	2, 4, 33	mp3an12 1480	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑦) = ((𝑆‘𝑦) +ℎ (𝑇‘𝑦)))
35	34	oveq2d 7426	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))))
36	35	adantl 486	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))))
37	1	lnopmuli 32324	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·ℎ 𝑦)) = (𝑥 ·ℎ (𝑆‘𝑦)))
38	3	lnopmuli 32324	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·ℎ 𝑦)) = (𝑥 ·ℎ (𝑇‘𝑦)))
39	37, 38	oveq12d 7428	. . . . . . 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 2808	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = ((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘(𝑥 ·ℎ 𝑦))))
41		hosval 32092	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑧) = ((𝑆‘𝑧) +ℎ (𝑇‘𝑧)))
42	2, 4, 41	mp3an12 1480	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑧) = ((𝑆‘𝑧) +ℎ (𝑇‘𝑧)))
43	40, 42	oveqan12d 7429	. . . . 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 2808	. . . 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 3157	. . 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 3205	. 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 32210	. 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 723	1 ⊢ (𝑆 +op 𝑇) ∈ LinOp

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℂcc 11093   ℋchba 31271   +_ℎ cva 31272   ·_ℎ csm 31273   +_op chos 31290  LinOpclo 31299

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 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-hilex 31351  ax-hfvadd 31352  ax-hvcom 31353  ax-hvass 31354  ax-hv0cl 31355  ax-hvaddid 31356  ax-hfvmul 31357  ax-hvmulid 31358  ax-hvdistr1 31360  ax-hvdistr2 31361  ax-hvmul0 31362

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-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-po 5569  df-so 5570  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-ltxr 11243  df-sub 11438  df-neg 11439  df-hvsub 31323  df-hosum 32082  df-lnop 32193

This theorem is referenced by:  lnophdi  32354  bdophsi  32448