Theorem lnophsi 32293

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 32261	. . 3 ⊢ 𝑆: ℋ⟶ ℋ
3		lnopco.2	. . . 4 ⊢ 𝑇 ∈ LinOp
4	3	lnopfi 32261	. . 3 ⊢ 𝑇: ℋ⟶ ℋ
5	2, 4	hoaddcli 32060	. 2 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
6		hvmulcl 31305	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ)
7	1	lnopaddi 32263	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑆‘𝑧)))
8	3	lnopaddi 32263	. . . . . . . 8 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑇‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘𝑧)))
9	7, 8	oveq12d 7429	. . . . . . 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 7079	. . . . . . . . 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 7079	. . . . . . . 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 7079	. . . . . . . . 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 7079	. . . . . . . 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 31328	. . . . . . 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 2804	. . . . 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 31304	. . . . . . 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 32032	. . . . . . 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 1477	. . . . . 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 7079	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑆‘𝑦) ∈ ℋ)
28	4	ffvelcdmi 7079	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (𝑇‘𝑦) ∈ ℋ)
29	27, 28	jca 520	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → ((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
30		ax-hvdistr1 31300	. . . . . . . . 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 1136	. . . . . . . 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 32032	. . . . . . . . . 10 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑦) = ((𝑆‘𝑦) +ℎ (𝑇‘𝑦)))
34	2, 4, 33	mp3an12 1477	. . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑦) = ((𝑆‘𝑦) +ℎ (𝑇‘𝑦)))
35	34	oveq2d 7427	. . . . . . . 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 32264	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑆‘(𝑥 ·ℎ 𝑦)) = (𝑥 ·ℎ (𝑆‘𝑦)))
38	3	lnopmuli 32264	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑇‘(𝑥 ·ℎ 𝑦)) = (𝑥 ·ℎ (𝑇‘𝑦)))
39	37, 38	oveq12d 7429	. . . . . . 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 2814	. . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ ((𝑆 +op 𝑇)‘𝑦)) = ((𝑆‘(𝑥 ·ℎ 𝑦)) +ℎ (𝑇‘(𝑥 ·ℎ 𝑦))))
41		hosval 32032	. . . . . . 7 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑧) = ((𝑆‘𝑧) +ℎ (𝑇‘𝑧)))
42	2, 4, 41	mp3an12 1477	. . . . . 6 ⊢ (𝑧 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑧) = ((𝑆‘𝑧) +ℎ (𝑇‘𝑧)))
43	40, 42	oveqan12d 7430	. . . . 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 2814	. . . 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 3163	. . 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 3211	. 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 32150	. 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 1567   ∈ wcel 2149  ∀wral 3085  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ℂcc 11097   ℋchba 31211   +_ℎ cva 31212   ·_ℎ csm 31213   +_op chos 31230  LinOpclo 31239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-hilex 31291  ax-hfvadd 31292  ax-hvcom 31293  ax-hvass 31294  ax-hv0cl 31295  ax-hvaddid 31296  ax-hfvmul 31297  ax-hvmulid 31298  ax-hvdistr1 31300  ax-hvdistr2 31301  ax-hvmul0 31302

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-po 5570  df-so 5571  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11244  df-mnf 11245  df-ltxr 11247  df-sub 11442  df-neg 11443  df-hvsub 31263  df-hosum 32022  df-lnop 32133

This theorem is referenced by:  lnophdi  32294  bdophsi  32388