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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressnm | Structured version Visualization version GIF version |
Description: The norm in a restricted structure. (Contributed by Thierry Arnoux, 8-Oct-2017.) |
Ref | Expression |
---|---|
ressnm.1 | ⊢ 𝐻 = (𝐺 ↾s 𝐴) |
ressnm.2 | ⊢ 𝐵 = (Base‘𝐺) |
ressnm.3 | ⊢ 0 = (0g‘𝐺) |
ressnm.4 | ⊢ 𝑁 = (norm‘𝐺) |
Ref | Expression |
---|---|
ressnm | ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝑁 ↾ 𝐴) = (norm‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressnm.1 | . . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝐴) | |
2 | ressnm.2 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 1, 2 | ressbas2 17221 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝐻)) |
4 | 3 | 3ad2ant3 1132 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 = (Base‘𝐻)) |
5 | 2 | fvexi 6910 | . . . . . . 7 ⊢ 𝐵 ∈ V |
6 | 5 | ssex 5322 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) |
7 | eqid 2725 | . . . . . . 7 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
8 | 1, 7 | ressds 17394 | . . . . . 6 ⊢ (𝐴 ∈ V → (dist‘𝐺) = (dist‘𝐻)) |
9 | 6, 8 | syl 17 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (dist‘𝐺) = (dist‘𝐻)) |
10 | 9 | 3ad2ant3 1132 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (dist‘𝐺) = (dist‘𝐻)) |
11 | eqidd 2726 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑥 = 𝑥) | |
12 | ressnm.3 | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
13 | 1, 2, 12 | ress0g 18725 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 0 = (0g‘𝐻)) |
14 | 10, 11, 13 | oveq123d 7440 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝑥(dist‘𝐺) 0 ) = (𝑥(dist‘𝐻)(0g‘𝐻))) |
15 | 4, 14 | mpteq12dv 5240 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝑥(dist‘𝐺) 0 )) = (𝑥 ∈ (Base‘𝐻) ↦ (𝑥(dist‘𝐻)(0g‘𝐻)))) |
16 | ressnm.4 | . . . . . 6 ⊢ 𝑁 = (norm‘𝐺) | |
17 | 16, 2, 12, 7 | nmfval 24541 | . . . . 5 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (𝑥(dist‘𝐺) 0 )) |
18 | 17 | reseq1i 5981 | . . . 4 ⊢ (𝑁 ↾ 𝐴) = ((𝑥 ∈ 𝐵 ↦ (𝑥(dist‘𝐺) 0 )) ↾ 𝐴) |
19 | resmpt 6042 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥(dist‘𝐺) 0 )) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝑥(dist‘𝐺) 0 ))) | |
20 | 18, 19 | eqtrid 2777 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑁 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝑥(dist‘𝐺) 0 ))) |
21 | 20 | 3ad2ant3 1132 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝑁 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝑥(dist‘𝐺) 0 ))) |
22 | eqid 2725 | . . . 4 ⊢ (norm‘𝐻) = (norm‘𝐻) | |
23 | eqid 2725 | . . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
24 | eqid 2725 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
25 | eqid 2725 | . . . 4 ⊢ (dist‘𝐻) = (dist‘𝐻) | |
26 | 22, 23, 24, 25 | nmfval 24541 | . . 3 ⊢ (norm‘𝐻) = (𝑥 ∈ (Base‘𝐻) ↦ (𝑥(dist‘𝐻)(0g‘𝐻))) |
27 | 26 | a1i 11 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (norm‘𝐻) = (𝑥 ∈ (Base‘𝐻) ↦ (𝑥(dist‘𝐻)(0g‘𝐻)))) |
28 | 15, 21, 27 | 3eqtr4d 2775 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝑁 ↾ 𝐴) = (norm‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ⊆ wss 3944 ↦ cmpt 5232 ↾ cres 5680 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 ↾s cress 17212 distcds 17245 0gc0g 17424 Mndcmnd 18697 normcnm 24529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-ds 17258 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-nm 24535 |
This theorem is referenced by: zringnm 33687 rezh 33700 |
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