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Mirrors > Home > MPE Home > Th. List > nmo0 | Structured version Visualization version GIF version |
Description: The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.) |
Ref | Expression |
---|---|
nmo0.1 | ⊢ 𝑁 = (𝑆 normOp 𝑇) |
nmo0.2 | ⊢ 𝑉 = (Base‘𝑆) |
nmo0.3 | ⊢ 0 = (0g‘𝑇) |
Ref | Expression |
---|---|
nmo0 | ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmo0.1 | . . 3 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
2 | nmo0.2 | . . 3 ⊢ 𝑉 = (Base‘𝑆) | |
3 | eqid 2821 | . . 3 ⊢ (norm‘𝑆) = (norm‘𝑆) | |
4 | eqid 2821 | . . 3 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
5 | eqid 2821 | . . 3 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
6 | simpl 485 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑆 ∈ NrmGrp) | |
7 | simpr 487 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑇 ∈ NrmGrp) | |
8 | ngpgrp 23208 | . . . 4 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp) | |
9 | ngpgrp 23208 | . . . 4 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp) | |
10 | nmo0.3 | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
11 | 10, 2 | 0ghm 18372 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑉 × { 0 }) ∈ (𝑆 GrpHom 𝑇)) |
12 | 8, 9, 11 | syl2an 597 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 GrpHom 𝑇)) |
13 | 0red 10644 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 0 ∈ ℝ) | |
14 | 0le0 11739 | . . . 4 ⊢ 0 ≤ 0 | |
15 | 14 | a1i 11 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 0 ≤ 0) |
16 | 10 | fvexi 6684 | . . . . . . . 8 ⊢ 0 ∈ V |
17 | 16 | fvconst2 6966 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑉 → ((𝑉 × { 0 })‘𝑥) = 0 ) |
18 | 17 | ad2antrl 726 | . . . . . 6 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝑉 × { 0 })‘𝑥) = 0 ) |
19 | 18 | fveq2d 6674 | . . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘((𝑉 × { 0 })‘𝑥)) = ((norm‘𝑇)‘ 0 )) |
20 | 4, 10 | nm0 23238 | . . . . . 6 ⊢ (𝑇 ∈ NrmGrp → ((norm‘𝑇)‘ 0 ) = 0) |
21 | 20 | ad2antlr 725 | . . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘ 0 ) = 0) |
22 | 19, 21 | eqtrd 2856 | . . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘((𝑉 × { 0 })‘𝑥)) = 0) |
23 | 2, 3 | nmcl 23225 | . . . . . . . 8 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → ((norm‘𝑆)‘𝑥) ∈ ℝ) |
24 | 23 | ad2ant2r 745 | . . . . . . 7 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℝ) |
25 | 24 | recnd 10669 | . . . . . 6 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ) |
26 | 25 | mul02d 10838 | . . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → (0 · ((norm‘𝑆)‘𝑥)) = 0) |
27 | 14, 26 | breqtrrid 5104 | . . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → 0 ≤ (0 · ((norm‘𝑆)‘𝑥))) |
28 | 22, 27 | eqbrtrd 5088 | . . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘((𝑉 × { 0 })‘𝑥)) ≤ (0 · ((norm‘𝑆)‘𝑥))) |
29 | 1, 2, 3, 4, 5, 6, 7, 12, 13, 15, 28 | nmolb2d 23327 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) ≤ 0) |
30 | 1 | nmoge0 23330 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ (𝑉 × { 0 }) ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘(𝑉 × { 0 }))) |
31 | 12, 30 | mpd3an3 1458 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 0 ≤ (𝑁‘(𝑉 × { 0 }))) |
32 | 1 | nmocl 23329 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ (𝑉 × { 0 }) ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘(𝑉 × { 0 })) ∈ ℝ*) |
33 | 12, 32 | mpd3an3 1458 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) ∈ ℝ*) |
34 | 0xr 10688 | . . 3 ⊢ 0 ∈ ℝ* | |
35 | xrletri3 12548 | . . 3 ⊢ (((𝑁‘(𝑉 × { 0 })) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑁‘(𝑉 × { 0 })) = 0 ↔ ((𝑁‘(𝑉 × { 0 })) ≤ 0 ∧ 0 ≤ (𝑁‘(𝑉 × { 0 }))))) | |
36 | 33, 34, 35 | sylancl 588 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ((𝑁‘(𝑉 × { 0 })) = 0 ↔ ((𝑁‘(𝑉 × { 0 })) ≤ 0 ∧ 0 ≤ (𝑁‘(𝑉 × { 0 }))))) |
37 | 29, 31, 36 | mpbir2and 711 | 1 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 {csn 4567 class class class wbr 5066 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 · cmul 10542 ℝ*cxr 10674 ≤ cle 10676 Basecbs 16483 0gc0g 16713 Grpcgrp 18103 GrpHom cghm 18355 normcnm 23186 NrmGrpcngp 23187 normOp cnmo 23314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ico 12745 df-0g 16715 df-topgen 16717 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-ghm 18356 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-xms 22930 df-ms 22931 df-nm 23192 df-ngp 23193 df-nmo 23317 |
This theorem is referenced by: nmoeq0 23345 0nghm 23350 idnghm 23352 |
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