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| Mirrors > Home > MPE Home > Th. List > nmods | Structured version Visualization version GIF version | ||
| Description: Upper bound for the distance between the values of a bounded linear operator. (Contributed by Mario Carneiro, 22-Oct-2015.) |
| Ref | Expression |
|---|---|
| nmods.n | ⊢ 𝑁 = (𝑆 normOp 𝑇) |
| nmods.v | ⊢ 𝑉 = (Base‘𝑆) |
| nmods.c | ⊢ 𝐶 = (dist‘𝑆) |
| nmods.d | ⊢ 𝐷 = (dist‘𝑇) |
| Ref | Expression |
|---|---|
| nmods | ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) ≤ ((𝑁‘𝐹) · (𝐴𝐶𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1137 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ (𝑆 NGHom 𝑇)) | |
| 2 | nghmrcl1 24753 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp) | |
| 3 | ngpgrp 24612 | . . . . 5 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp) | |
| 4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ Grp) |
| 5 | nmods.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑆) | |
| 6 | eqid 2737 | . . . . 5 ⊢ (-g‘𝑆) = (-g‘𝑆) | |
| 7 | 5, 6 | grpsubcl 19038 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(-g‘𝑆)𝐵) ∈ 𝑉) |
| 8 | 4, 7 | syl3an1 1164 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(-g‘𝑆)𝐵) ∈ 𝑉) |
| 9 | nmods.n | . . . 4 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
| 10 | eqid 2737 | . . . 4 ⊢ (norm‘𝑆) = (norm‘𝑆) | |
| 11 | eqid 2737 | . . . 4 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
| 12 | 9, 5, 10, 11 | nmoi 24749 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝐴(-g‘𝑆)𝐵) ∈ 𝑉) → ((norm‘𝑇)‘(𝐹‘(𝐴(-g‘𝑆)𝐵))) ≤ ((𝑁‘𝐹) · ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵)))) |
| 13 | 1, 8, 12 | syl2anc 584 | . 2 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑇)‘(𝐹‘(𝐴(-g‘𝑆)𝐵))) ≤ ((𝑁‘𝐹) · ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵)))) |
| 14 | nghmrcl2 24754 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp) | |
| 15 | 14 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ NrmGrp) |
| 16 | nghmghm 24755 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
| 17 | 16 | 3ad2ant1 1134 | . . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
| 18 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
| 19 | 5, 18 | ghmf 19238 | . . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑉⟶(Base‘𝑇)) |
| 20 | 17, 19 | syl 17 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐹:𝑉⟶(Base‘𝑇)) |
| 21 | simp2 1138 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
| 22 | 20, 21 | ffvelcdmd 7105 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝐴) ∈ (Base‘𝑇)) |
| 23 | simp3 1139 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
| 24 | 20, 23 | ffvelcdmd 7105 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝐵) ∈ (Base‘𝑇)) |
| 25 | eqid 2737 | . . . . 5 ⊢ (-g‘𝑇) = (-g‘𝑇) | |
| 26 | nmods.d | . . . . 5 ⊢ 𝐷 = (dist‘𝑇) | |
| 27 | 11, 18, 25, 26 | ngpds 24617 | . . . 4 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝐴) ∈ (Base‘𝑇) ∧ (𝐹‘𝐵) ∈ (Base‘𝑇)) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) = ((norm‘𝑇)‘((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵)))) |
| 28 | 15, 22, 24, 27 | syl3anc 1373 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) = ((norm‘𝑇)‘((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵)))) |
| 29 | 5, 6, 25 | ghmsub 19242 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘(𝐴(-g‘𝑆)𝐵)) = ((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵))) |
| 30 | 16, 29 | syl3an1 1164 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘(𝐴(-g‘𝑆)𝐵)) = ((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵))) |
| 31 | 30 | fveq2d 6910 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑇)‘(𝐹‘(𝐴(-g‘𝑆)𝐵))) = ((norm‘𝑇)‘((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵)))) |
| 32 | 28, 31 | eqtr4d 2780 | . 2 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) = ((norm‘𝑇)‘(𝐹‘(𝐴(-g‘𝑆)𝐵)))) |
| 33 | nmods.c | . . . . 5 ⊢ 𝐶 = (dist‘𝑆) | |
| 34 | 10, 5, 6, 33 | ngpds 24617 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴𝐶𝐵) = ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵))) |
| 35 | 2, 34 | syl3an1 1164 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴𝐶𝐵) = ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵))) |
| 36 | 35 | oveq2d 7447 | . 2 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑁‘𝐹) · (𝐴𝐶𝐵)) = ((𝑁‘𝐹) · ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵)))) |
| 37 | 13, 32, 36 | 3brtr4d 5175 | 1 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) ≤ ((𝑁‘𝐹) · (𝐴𝐶𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 · cmul 11160 ≤ cle 11296 Basecbs 17247 distcds 17306 Grpcgrp 18951 -gcsg 18953 GrpHom cghm 19230 normcnm 24589 NrmGrpcngp 24590 normOp cnmo 24726 NGHom cnghm 24727 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-0g 17486 df-topgen 17488 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-ghm 19231 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-xms 24330 df-ms 24331 df-nm 24595 df-ngp 24596 df-nmo 24729 df-nghm 24730 |
| This theorem is referenced by: nghmcn 24766 |
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