Theorem nmoptrii 32389

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 32157	. . . . 5 ⊢ (𝑆 ∈ BndLinOp → 𝑆: ℋ⟶ ℋ)
3	1, 2	ax-mp 5	. . . 4 ⊢ 𝑆: ℋ⟶ ℋ
4		nmoptri.2	. . . . 5 ⊢ 𝑇 ∈ BndLinOp
5		bdopf 32157	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
6	4, 5	ax-mp 5	. . . 4 ⊢ 𝑇: ℋ⟶ ℋ
7	3, 6	hoaddcli 32063	. . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
8		nmopre 32165	. . . . . 6 ⊢ (𝑆 ∈ BndLinOp → (normop‘𝑆) ∈ ℝ)
9	1, 8	ax-mp 5	. . . . 5 ⊢ (normop‘𝑆) ∈ ℝ
10		nmopre 32165	. . . . . 6 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ)
11	4, 10	ax-mp 5	. . . . 5 ⊢ (normop‘𝑇) ∈ ℝ
12	9, 11	readdcli 11226	. . . 4 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ
13	12	rexri 11269	. . 3 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ*
14		nmopub 32203	. . 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 704	. 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 32057	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) ∈ ℋ)
17		normcl 31420	. . . . . 6 ⊢ (((𝑆 +op 𝑇)‘𝑥) ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ∈ ℝ)
18	16, 17	syl 18	. . . . 5 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ∈ ℝ)
19	18	adantr 485	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ∈ ℝ)
20	3	ffvelcdmi 7081	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ)
21		normcl 31420	. . . . . . 7 ⊢ ((𝑆‘𝑥) ∈ ℋ → (normℎ‘(𝑆‘𝑥)) ∈ ℝ)
22	20, 21	syl 18	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑆‘𝑥)) ∈ ℝ)
23	6	ffvelcdmi 7081	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
24		normcl 31420	. . . . . . 7 ⊢ ((𝑇‘𝑥) ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
25	23, 24	syl 18	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
26	22, 25	readdcld 11240	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ∈ ℝ)
27	26	adantr 485	. . . 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 32035	. . . . . . . 8 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
30	3, 6, 29	mp3an12 1477	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
31	30	fveq2d 6888	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) = (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
32		norm-ii 31433	. . . . . . 7 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
33	20, 23, 32	syl2anc 595	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
34	31, 33	eqbrtrd 5137	. . . . 5 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
35	34	adantr 485	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
36		nmoplb 32202	. . . . . 6 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆))
37	3, 36	mp3an1 1474	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆))
38		nmoplb 32202	. . . . . 6 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
39	6, 38	mp3an1 1474	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
40		le2add 11698	. . . . . . . 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 717	. . . . . . 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 595	. . . . . 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 485	. . . . 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 711	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ≤ ((normop‘𝑆) + (normop‘𝑇)))
45	19, 27, 28, 35, 44	letrd 11369	. . 3 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normop‘𝑆) + (normop‘𝑇)))
46	45	ex 417	. 2 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) ≤ 1 → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normop‘𝑆) + (normop‘𝑇))))
47	15, 46	mprgbir 3092	1 ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   class class class wbr 5113  ⟶wf 6535  ‘cfv 6539  (class class class)co 7413  ℝcr 11101  1c1 11103   + caddc 11105  ℝ^*cxr 11244   ≤ cle 11246   ℋchba 31214   +_ℎ cva 31215  norm_ℎcno 31218   +_op chos 31233  norm_opcnop 31240  BndLinOpcbo 31243

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 5273  ax-pow 5339  ax-pr 5407  ax-un 7735  ax-cnex 11158  ax-resscn 11159  ax-1cn 11160  ax-icn 11161  ax-addcl 11162  ax-addrcl 11163  ax-mulcl 11164  ax-mulrcl 11165  ax-mulcom 11166  ax-addass 11167  ax-mulass 11168  ax-distr 11169  ax-i2m1 11170  ax-1ne0 11171  ax-1rid 11172  ax-rnegex 11173  ax-rrecex 11174  ax-cnre 11175  ax-pre-lttri 11176  ax-pre-lttrn 11177  ax-pre-ltadd 11178  ax-pre-mulgt0 11179  ax-pre-sup 11180  ax-hilex 31294  ax-hfvadd 31295  ax-hvcom 31296  ax-hvass 31297  ax-hv0cl 31298  ax-hvaddid 31299  ax-hfvmul 31300  ax-hvmulid 31301  ax-hvmulass 31302  ax-hvdistr1 31303  ax-hvdistr2 31304  ax-hvmul0 31305  ax-hfi 31374  ax-his1 31377  ax-his2 31378  ax-his3 31379  ax-his4 31380

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-rmo 3376  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-pss 3933  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-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-sup 9404  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11445  df-neg 11446  df-div 11874  df-nn 12236  df-2 12305  df-3 12306  df-4 12307  df-n0 12507  df-z 12594  df-uz 12865  df-rp 13019  df-seq 14040  df-exp 14100  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-grpo 30788  df-gid 30789  df-ablo 30840  df-vc 30854  df-nv 30887  df-va 30890  df-ba 30891  df-sm 30892  df-0v 30893  df-nmcv 30895  df-hnorm 31263  df-hba 31264  df-hvsub 31266  df-hosum 32025  df-nmop 32134  df-lnop 32136  df-bdop 32137

This theorem is referenced by:  bdophsi  32391  nmoptri2i  32394  unierri  32399