Theorem nmoptrii 30175

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 29943	. . . . 5 ⊢ (𝑆 ∈ BndLinOp → 𝑆: ℋ⟶ ℋ)
3	1, 2	ax-mp 5	. . . 4 ⊢ 𝑆: ℋ⟶ ℋ
4		nmoptri.2	. . . . 5 ⊢ 𝑇 ∈ BndLinOp
5		bdopf 29943	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
6	4, 5	ax-mp 5	. . . 4 ⊢ 𝑇: ℋ⟶ ℋ
7	3, 6	hoaddcli 29849	. . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
8		nmopre 29951	. . . . . 6 ⊢ (𝑆 ∈ BndLinOp → (normop‘𝑆) ∈ ℝ)
9	1, 8	ax-mp 5	. . . . 5 ⊢ (normop‘𝑆) ∈ ℝ
10		nmopre 29951	. . . . . 6 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ)
11	4, 10	ax-mp 5	. . . . 5 ⊢ (normop‘𝑇) ∈ ℝ
12	9, 11	readdcli 10848	. . . 4 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ
13	12	rexri 10891	. . 3 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ*
14		nmopub 29989	. . 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 29843	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) ∈ ℋ)
17		normcl 29206	. . . . . 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 484	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ∈ ℝ)
20	3	ffvelrni 6903	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ)
21		normcl 29206	. . . . . . 7 ⊢ ((𝑆‘𝑥) ∈ ℋ → (normℎ‘(𝑆‘𝑥)) ∈ ℝ)
22	20, 21	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑆‘𝑥)) ∈ ℝ)
23	6	ffvelrni 6903	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
24		normcl 29206	. . . . . . 7 ⊢ ((𝑇‘𝑥) ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
25	23, 24	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
26	22, 25	readdcld 10862	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ∈ ℝ)
27	26	adantr 484	. . . 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 29821	. . . . . . . 8 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
30	3, 6, 29	mp3an12 1453	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
31	30	fveq2d 6721	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) = (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
32		norm-ii 29219	. . . . . . 7 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
33	20, 23, 32	syl2anc 587	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
34	31, 33	eqbrtrd 5075	. . . . 5 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
35	34	adantr 484	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
36		nmoplb 29988	. . . . . 6 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆))
37	3, 36	mp3an1 1450	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆))
38		nmoplb 29988	. . . . . 6 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
39	6, 38	mp3an1 1450	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
40		le2add 11314	. . . . . . . 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 587	. . . . . 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 484	. . . . 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 10989	. . 3 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normop‘𝑆) + (normop‘𝑇)))
46	45	ex 416	. 2 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) ≤ 1 → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normop‘𝑆) + (normop‘𝑇))))
47	15, 46	mprgbir 3076	1 ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061   class class class wbr 5053  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213  ℝcr 10728  1c1 10730   + caddc 10732  ℝ^*cxr 10866   ≤ cle 10868   ℋchba 29000   +_ℎ cva 29001  norm_ℎcno 29004   +_op chos 29019  norm_opcnop 29026  BndLinOpcbo 29029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-hilex 29080  ax-hfvadd 29081  ax-hvcom 29082  ax-hvass 29083  ax-hv0cl 29084  ax-hvaddid 29085  ax-hfvmul 29086  ax-hvmulid 29087  ax-hvmulass 29088  ax-hvdistr1 29089  ax-hvdistr2 29090  ax-hvmul0 29091  ax-hfi 29160  ax-his1 29163  ax-his2 29164  ax-his3 29165  ax-his4 29166

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-sup 9058  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-n0 12091  df-z 12177  df-uz 12439  df-rp 12587  df-seq 13575  df-exp 13636  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-grpo 28574  df-gid 28575  df-ablo 28626  df-vc 28640  df-nv 28673  df-va 28676  df-ba 28677  df-sm 28678  df-0v 28679  df-nmcv 28681  df-hnorm 29049  df-hba 29050  df-hvsub 29052  df-hosum 29811  df-nmop 29920  df-lnop 29922  df-bdop 29923

This theorem is referenced by:  bdophsi  30177  nmoptri2i  30180  unierri  30185