Theorem lnophsi 32201

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 32169	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 32169	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hoaddcli 31968	. 2 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
6		hvmulcl 31213	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
7	1	lnopaddi 32171	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑆‘𝑧)))
8	3	lnopaddi 32171	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑇‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘𝑧)))
9	7, 8	oveq12d 7414	. . . . . . 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 589	. . . . . 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 7064	. . . . . . . . 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 7064	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑆‘𝑧) ∈ ℋ)
14	12, 13	anim12i 622	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆‘(𝑥 ·ℎ 𝑦)) ∈ ℋ ∧ (𝑆‘𝑧) ∈ ℋ))
15	4	ffvelcdmi 7064	. . . . . . . . 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 7064	. . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ)
18	16, 17	anim12i 622	. . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑇‘(𝑥 ·ℎ 𝑦)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ))
19		hvadd4 31236	. . . . . . 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 593	. . . . . 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 2797	. . . . 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 31212	. . . . . . 7 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
23	6, 22	sylan 589	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ)
24		hosval 31940	. . . . . . 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 1472	. . . . . 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 7064	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑆‘𝑦) ∈ ℋ)
28	4	ffvelcdmi 7064	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
29	27, 28	jca 519	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
30		ax-hvdistr1 31208	. . . . . . . . 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 1133	. . . . . . . 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 602	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))) = ((𝑥 ·ℎ (𝑆‘𝑦)) +ℎ (𝑥 ·ℎ (𝑇‘𝑦))))
33		hosval 31940	. . . . . . . . . 10 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑦) = ((𝑆‘𝑦) +ℎ (𝑇‘𝑦)))
34	2, 4, 33	mp3an12 1472	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑦) = ((𝑆‘𝑦) +ℎ (𝑇‘𝑦)))
35	34	oveq2d 7412	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))))
36	35	adantl 485	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = (𝑥 ·ℎ ((𝑆‘𝑦) +ℎ (𝑇‘𝑦))))
37	1	lnopmuli 32172	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·ℎ 𝑦)) = (𝑥 ·ℎ (𝑆‘𝑦)))
38	3	lnopmuli 32172	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·ℎ 𝑦)) = (𝑥 ·ℎ (𝑇‘𝑦)))
39	37, 38	oveq12d 7414	. . . . . . 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 2807	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = ((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘(𝑥 ·ℎ 𝑦))))
41		hosval 31940	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑧) = ((𝑆‘𝑧) +ℎ (𝑇‘𝑧)))
42	2, 4, 41	mp3an12 1472	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑧) = ((𝑆‘𝑧) +ℎ (𝑇‘𝑧)))
43	40, 42	oveqan12d 7415	. . . . 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 2807	. . . 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 3154	. . 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 3202	. 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 32058	. 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 721	1 ⊢ (𝑆 +op 𝑇) ∈ LinOp

Syntax hints:   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  ℂcc 11071   ℋchba 31119   +_ℎ cva 31120   ·_ℎ csm 31121   +_op chos 31138  LinOpclo 31147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-hilex 31199  ax-hfvadd 31200  ax-hvcom 31201  ax-hvass 31202  ax-hv0cl 31203  ax-hvaddid 31204  ax-hfvmul 31205  ax-hvmulid 31206  ax-hvdistr1 31208  ax-hvdistr2 31209  ax-hvmul0 31210

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-po 5555  df-so 5556  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-er 8678  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11218  df-mnf 11219  df-ltxr 11221  df-sub 11416  df-neg 11417  df-hvsub 31171  df-hosum 31930  df-lnop 32041

This theorem is referenced by:  lnophdi  32202  bdophsi  32296