Theorem nmoptrii 32060

Description: Triangle inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nmoptri.1	⊢ 𝑆 ∈ BndLinOp
nmoptri.2	⊢ 𝑇 ∈ BndLinOp

Assertion

Ref	Expression
nmoptrii	⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇))

Proof of Theorem nmoptrii

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmoptri.1	. . . . 5 ⊢ 𝑆 ∈ BndLinOp
2		bdopf 31828	. . . . 5 ⊢ (𝑆 ∈ BndLinOp → 𝑆: ℋ⟶ ℋ)
3	1, 2	ax-mp 5	. . . 4 ⊢ 𝑆: ℋ⟶ ℋ
4		nmoptri.2	. . . . 5 ⊢ 𝑇 ∈ BndLinOp
5		bdopf 31828	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
6	4, 5	ax-mp 5	. . . 4 ⊢ 𝑇: ℋ⟶ ℋ
7	3, 6	hoaddcli 31734	. . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
8		nmopre 31836	. . . . . 6 ⊢ (𝑆 ∈ BndLinOp → (normop‘𝑆) ∈ ℝ)
9	1, 8	ax-mp 5	. . . . 5 ⊢ (normop‘𝑆) ∈ ℝ
10		nmopre 31836	. . . . . 6 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ)
11	4, 10	ax-mp 5	. . . . 5 ⊢ (normop‘𝑇) ∈ ℝ
12	9, 11	readdcli 11259	. . . 4 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ
13	12	rexri 11302	. . 3 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ*
14		nmopub 31874	. . 3 ⊢ (((𝑆 +op 𝑇): ℋ⟶ ℋ ∧ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ*) → ((normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)) ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normop‘𝑆) + (normop‘𝑇)))))
15	7, 13, 14	mp2an 692	. 2 ⊢ ((normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)) ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normop‘𝑆) + (normop‘𝑇))))
16	3, 6	hoscli 31728	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) ∈ ℋ)
17		normcl 31091	. . . . . 6 ⊢ (((𝑆 +op 𝑇)‘𝑥) ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ∈ ℝ)
18	16, 17	syl 17	. . . . 5 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ∈ ℝ)
19	18	adantr 480	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ∈ ℝ)
20	3	ffvelcdmi 7084	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ)
21		normcl 31091	. . . . . . 7 ⊢ ((𝑆‘𝑥) ∈ ℋ → (normℎ‘(𝑆‘𝑥)) ∈ ℝ)
22	20, 21	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑆‘𝑥)) ∈ ℝ)
23	6	ffvelcdmi 7084	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
24		normcl 31091	. . . . . . 7 ⊢ ((𝑇‘𝑥) ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
25	23, 24	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
26	22, 25	readdcld 11273	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ∈ ℝ)
27	26	adantr 480	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ∈ ℝ)
28	12	a1i 11	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ)
29		hosval 31706	. . . . . . . 8 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
30	3, 6, 29	mp3an12 1452	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
31	30	fveq2d 6891	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) = (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
32		norm-ii 31104	. . . . . . 7 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
33	20, 23, 32	syl2anc 584	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
34	31, 33	eqbrtrd 5147	. . . . 5 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
35	34	adantr 480	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
36		nmoplb 31873	. . . . . 6 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆))
37	3, 36	mp3an1 1449	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆))
38		nmoplb 31873	. . . . . 6 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
39	6, 38	mp3an1 1449	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
40		le2add 11728	. . . . . . . 8 ⊢ ((((normℎ‘(𝑆‘𝑥)) ∈ ℝ ∧ (normℎ‘(𝑇‘𝑥)) ∈ ℝ) ∧ ((normop‘𝑆) ∈ ℝ ∧ (normop‘𝑇) ∈ ℝ)) → (((normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆) ∧ (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇)) → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ≤ ((normop‘𝑆) + (normop‘𝑇))))
41	9, 11, 40	mpanr12 705	. . . . . . 7 ⊢ (((normℎ‘(𝑆‘𝑥)) ∈ ℝ ∧ (normℎ‘(𝑇‘𝑥)) ∈ ℝ) → (((normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆) ∧ (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇)) → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ≤ ((normop‘𝑆) + (normop‘𝑇))))
42	22, 25, 41	syl2anc 584	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (((normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆) ∧ (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇)) → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ≤ ((normop‘𝑆) + (normop‘𝑇))))
43	42	adantr 480	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (((normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆) ∧ (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇)) → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ≤ ((normop‘𝑆) + (normop‘𝑇))))
44	37, 39, 43	mp2and 699	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ≤ ((normop‘𝑆) + (normop‘𝑇)))
45	19, 27, 28, 35, 44	letrd 11401	. . 3 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normop‘𝑆) + (normop‘𝑇)))
46	45	ex 412	. 2 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) ≤ 1 → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normop‘𝑆) + (normop‘𝑇))))
47	15, 46	mprgbir 3057	1 ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050   class class class wbr 5125  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414  ℝcr 11137  1c1 11139   + caddc 11141  ℝ^*cxr 11277   ≤ cle 11279   ℋchba 30885   +_ℎ cva 30886  norm_ℎcno 30889   +_op chos 30904  norm_opcnop 30911  BndLinOpcbo 30914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216  ax-hilex 30965  ax-hfvadd 30966  ax-hvcom 30967  ax-hvass 30968  ax-hv0cl 30969  ax-hvaddid 30970  ax-hfvmul 30971  ax-hvmulid 30972  ax-hvmulass 30973  ax-hvdistr1 30974  ax-hvdistr2 30975  ax-hvmul0 30976  ax-hfi 31045  ax-his1 31048  ax-his2 31049  ax-his3 31050  ax-his4 31051

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7871  df-1st 7997  df-2nd 7998  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433  df-er 8728  df-map 8851  df-en 8969  df-dom 8970  df-sdom 8971  df-sup 9465  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11477  df-neg 11478  df-div 11904  df-nn 12250  df-2 12312  df-3 12313  df-4 12314  df-n0 12511  df-z 12598  df-uz 12862  df-rp 13018  df-seq 14026  df-exp 14086  df-cj 15121  df-re 15122  df-im 15123  df-sqrt 15257  df-abs 15258  df-grpo 30459  df-gid 30460  df-ablo 30511  df-vc 30525  df-nv 30558  df-va 30561  df-ba 30562  df-sm 30563  df-0v 30564  df-nmcv 30566  df-hnorm 30934  df-hba 30935  df-hvsub 30937  df-hosum 31696  df-nmop 31805  df-lnop 31807  df-bdop 31808

This theorem is referenced by:  bdophsi  32062  nmoptri2i  32065  unierri  32070