Theorem nmoptrii 32123

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 31891	. . . . 5 ⊢ (𝑆 ∈ BndLinOp → 𝑆: ℋ⟶ ℋ)
3	1, 2	ax-mp 5	. . . 4 ⊢ 𝑆: ℋ⟶ ℋ
4		nmoptri.2	. . . . 5 ⊢ 𝑇 ∈ BndLinOp
5		bdopf 31891	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
6	4, 5	ax-mp 5	. . . 4 ⊢ 𝑇: ℋ⟶ ℋ
7	3, 6	hoaddcli 31797	. . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
8		nmopre 31899	. . . . . 6 ⊢ (𝑆 ∈ BndLinOp → (normop‘𝑆) ∈ ℝ)
9	1, 8	ax-mp 5	. . . . 5 ⊢ (normop‘𝑆) ∈ ℝ
10		nmopre 31899	. . . . . 6 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ)
11	4, 10	ax-mp 5	. . . . 5 ⊢ (normop‘𝑇) ∈ ℝ
12	9, 11	readdcli 11274	. . . 4 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ
13	12	rexri 11317	. . 3 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ*
14		nmopub 31937	. . 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 31791	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) ∈ ℋ)
17		normcl 31154	. . . . . 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 7103	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ)
21		normcl 31154	. . . . . . 7 ⊢ ((𝑆‘𝑥) ∈ ℋ → (normℎ‘(𝑆‘𝑥)) ∈ ℝ)
22	20, 21	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑆‘𝑥)) ∈ ℝ)
23	6	ffvelcdmi 7103	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
24		normcl 31154	. . . . . . 7 ⊢ ((𝑇‘𝑥) ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
25	23, 24	syl 17	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
26	22, 25	readdcld 11288	. . . . 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 31769	. . . . . . . 8 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
30	3, 6, 29	mp3an12 1450	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
31	30	fveq2d 6911	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) = (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
32		norm-ii 31167	. . . . . . 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 5170	. . . . 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 31936	. . . . . 6 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆))
37	3, 36	mp3an1 1447	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆))
38		nmoplb 31936	. . . . . 6 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
39	6, 38	mp3an1 1447	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
40		le2add 11743	. . . . . . . 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 11416	. . 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 3066	1 ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059   class class class wbr 5148  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  ℝcr 11152  1c1 11154   + caddc 11156  ℝ^*cxr 11292   ≤ cle 11294   ℋchba 30948   +_ℎ cva 30949  norm_ℎcno 30952   +_op chos 30967  norm_opcnop 30974  BndLinOpcbo 30977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-hilex 31028  ax-hfvadd 31029  ax-hvcom 31030  ax-hvass 31031  ax-hv0cl 31032  ax-hvaddid 31033  ax-hfvmul 31034  ax-hvmulid 31035  ax-hvmulass 31036  ax-hvdistr1 31037  ax-hvdistr2 31038  ax-hvmul0 31039  ax-hfi 31108  ax-his1 31111  ax-his2 31112  ax-his3 31113  ax-his4 31114

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-grpo 30522  df-gid 30523  df-ablo 30574  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-nmcv 30629  df-hnorm 30997  df-hba 30998  df-hvsub 31000  df-hosum 31759  df-nmop 31868  df-lnop 31870  df-bdop 31871

This theorem is referenced by:  bdophsi  32125  nmoptri2i  32128  unierri  32133