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 16538 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝐻)) |
4 | 3 | 3ad2ant3 1131 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝐴 = (Base‘𝐻)) |
5 | 2 | fvexi 6670 | . . . . . . 7 ⊢ 𝐵 ∈ V |
6 | 5 | ssex 5211 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) |
7 | eqid 2821 | . . . . . . 7 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
8 | 1, 7 | ressds 16669 | . . . . . 6 ⊢ (𝐴 ∈ V → (dist‘𝐺) = (dist‘𝐻)) |
9 | 6, 8 | syl 17 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (dist‘𝐺) = (dist‘𝐻)) |
10 | 9 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (dist‘𝐺) = (dist‘𝐻)) |
11 | eqidd 2822 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 𝑥 = 𝑥) | |
12 | ressnm.3 | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
13 | 1, 2, 12 | ress0g 17922 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 0 = (0g‘𝐻)) |
14 | 10, 11, 13 | oveq123d 7163 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝑥(dist‘𝐺) 0 ) = (𝑥(dist‘𝐻)(0g‘𝐻))) |
15 | 4, 14 | mpteq12dv 5137 | . 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 23181 | . . . . 5 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (𝑥(dist‘𝐺) 0 )) |
18 | 17 | reseq1i 5835 | . . . 4 ⊢ (𝑁 ↾ 𝐴) = ((𝑥 ∈ 𝐵 ↦ (𝑥(dist‘𝐺) 0 )) ↾ 𝐴) |
19 | resmpt 5891 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝑥(dist‘𝐺) 0 )) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝑥(dist‘𝐺) 0 ))) | |
20 | 18, 19 | syl5eq 2868 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑁 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝑥(dist‘𝐺) 0 ))) |
21 | 20 | 3ad2ant3 1131 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝑁 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝑥(dist‘𝐺) 0 ))) |
22 | eqid 2821 | . . . 4 ⊢ (norm‘𝐻) = (norm‘𝐻) | |
23 | eqid 2821 | . . . 4 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
24 | eqid 2821 | . . . 4 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
25 | eqid 2821 | . . . 4 ⊢ (dist‘𝐻) = (dist‘𝐻) | |
26 | 22, 23, 24, 25 | nmfval 23181 | . . 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 2866 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝑁 ↾ 𝐴) = (norm‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3486 ⊆ wss 3924 ↦ cmpt 5132 ↾ cres 5543 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 ↾s cress 16467 distcds 16557 0gc0g 16696 Mndcmnd 17894 normcnm 23169 |
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-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-ress 16474 df-plusg 16561 df-ds 16570 df-0g 16698 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-nm 23175 |
This theorem is referenced by: zringnm 31208 rezh 31219 |
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