Theorem nmoptrii 32711

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 32479	. . . . 5 ⊢ (𝑆 ∈ BndLinOp → 𝑆: ℋ⟶ ℋ)
3	1, 2	ax-mp 5	. . . 4 ⊢ 𝑆: ℋ⟶ ℋ
4		nmoptri.2	. . . . 5 ⊢ 𝑇 ∈ BndLinOp
5		bdopf 32479	. . . . 5 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)
6	4, 5	ax-mp 5	. . . 4 ⊢ 𝑇: ℋ⟶ ℋ
7	3, 6	hoaddcli 32385	. . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ
8		nmopre 32487	. . . . . 6 ⊢ (𝑆 ∈ BndLinOp → (normop‘𝑆) ∈ ℝ)
9	1, 8	ax-mp 5	. . . . 5 ⊢ (normop‘𝑆) ∈ ℝ
10		nmopre 32487	. . . . . 6 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ)
11	4, 10	ax-mp 5	. . . . 5 ⊢ (normop‘𝑇) ∈ ℝ
12	9, 11	readdcli 11337	. . . 4 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ
13	12	rexri 11380	. . 3 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ*
14		nmopub 32525	. . 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 705	. 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 32379	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) ∈ ℋ)
17		normcl 31742	. . . . . 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 486	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ∈ ℝ)
20	3	ffvelcdmi 7094	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ)
21		normcl 31742	. . . . . . 7 ⊢ ((𝑆‘𝑥) ∈ ℋ → (normℎ‘(𝑆‘𝑥)) ∈ ℝ)
22	20, 21	syl 18	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑆‘𝑥)) ∈ ℝ)
23	6	ffvelcdmi 7094	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ)
24		normcl 31742	. . . . . . 7 ⊢ ((𝑇‘𝑥) ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
25	23, 24	syl 18	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘(𝑇‘𝑥)) ∈ ℝ)
26	22, 25	readdcld 11351	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ∈ ℝ)
27	26	adantr 486	. . . 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 32357	. . . . . . . 8 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
30	3, 6, 29	mp3an12 1480	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)))
31	30	fveq2d 6900	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) = (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))))
32		norm-ii 31755	. . . . . . 7 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
33	20, 23, 32	syl2anc 596	. . . . . 6 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
34	31, 33	eqbrtrd 5130	. . . . 5 ⊢ (𝑥 ∈ ℋ → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
35	34	adantr 486	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))))
36		nmoplb 32524	. . . . . 6 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆))
37	3, 36	mp3an1 1477	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑆‘𝑥)) ≤ (normop‘𝑆))
38		nmoplb 32524	. . . . . 6 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
39	6, 38	mp3an1 1477	. . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇))
40		le2add 11811	. . . . . . . 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 718	. . . . . . 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 596	. . . . . 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 486	. . . . 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 712	. . . 4 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → ((normℎ‘(𝑆‘𝑥)) + (normℎ‘(𝑇‘𝑥))) ≤ ((normop‘𝑆) + (normop‘𝑇)))
45	19, 27, 28, 35, 44	letrd 11480	. . 3 ⊢ ((𝑥 ∈ ℋ ∧ (normℎ‘𝑥) ≤ 1) → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normop‘𝑆) + (normop‘𝑇)))
46	45	ex 418	. 2 ⊢ (𝑥 ∈ ℋ → ((normℎ‘𝑥) ≤ 1 → (normℎ‘((𝑆 +op 𝑇)‘𝑥)) ≤ ((normop‘𝑆) + (normop‘𝑇))))
47	15, 46	mprgbir 3087	1 ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∀wral 3080   class class class wbr 5106  ⟶wf 6543  ‘cfv 6547  (class class class)co 7430  ℝcr 11212  1c1 11214   + caddc 11216  ℝ^*cxr 11355   ≤ cle 11357   ℋchba 31536   +_ℎ cva 31537  norm_ℎcno 31540   +_op chos 31555  norm_opcnop 31562  BndLinOpcbo 31565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290  ax-pre-sup 11291  ax-hilex 31616  ax-hfvadd 31617  ax-hvcom 31618  ax-hvass 31619  ax-hv0cl 31620  ax-hvaddid 31621  ax-hfvmul 31622  ax-hvmulid 31623  ax-hvmulass 31624  ax-hvdistr1 31625  ax-hvdistr2 31626  ax-hvmul0 31627  ax-hfi 31696  ax-his1 31699  ax-his2 31700  ax-his3 31701  ax-his4 31702

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-er 8731  df-map 8863  df-en 8988  df-dom 8989  df-sdom 8990  df-sup 9447  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-div 11987  df-nn 12349  df-2 12418  df-3 12419  df-4 12420  df-n0 12620  df-z 12707  df-uz 12979  df-rp 13134  df-seq 14158  df-exp 14218  df-cj 15281  df-re 15282  df-im 15283  df-sqrt 15417  df-abs 15418  df-grpo 31110  df-gid 31111  df-ablo 31162  df-vc 31176  df-nv 31209  df-va 31212  df-ba 31213  df-sm 31214  df-0v 31215  df-nmcv 31217  df-hnorm 31585  df-hba 31586  df-hvsub 31588  df-hosum 32347  df-nmop 32456  df-lnop 32458  df-bdop 32459

This theorem is used by:  bdophsi  32713  nmoptri2i  32716  unierri  32721