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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 | β’ π» = (πΊ βΎs π΄) |
ressnm.2 | β’ π΅ = (BaseβπΊ) |
ressnm.3 | β’ 0 = (0gβπΊ) |
ressnm.4 | β’ π = (normβπΊ) |
Ref | Expression |
---|---|
ressnm | β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β (π βΎ π΄) = (normβπ»)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressnm.1 | . . . . 5 β’ π» = (πΊ βΎs π΄) | |
2 | ressnm.2 | . . . . 5 β’ π΅ = (BaseβπΊ) | |
3 | 1, 2 | ressbas2 17181 | . . . 4 β’ (π΄ β π΅ β π΄ = (Baseβπ»)) |
4 | 3 | 3ad2ant3 1135 | . . 3 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β π΄ = (Baseβπ»)) |
5 | 2 | fvexi 6905 | . . . . . . 7 β’ π΅ β V |
6 | 5 | ssex 5321 | . . . . . 6 β’ (π΄ β π΅ β π΄ β V) |
7 | eqid 2732 | . . . . . . 7 β’ (distβπΊ) = (distβπΊ) | |
8 | 1, 7 | ressds 17354 | . . . . . 6 β’ (π΄ β V β (distβπΊ) = (distβπ»)) |
9 | 6, 8 | syl 17 | . . . . 5 β’ (π΄ β π΅ β (distβπΊ) = (distβπ»)) |
10 | 9 | 3ad2ant3 1135 | . . . 4 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β (distβπΊ) = (distβπ»)) |
11 | eqidd 2733 | . . . 4 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β π₯ = π₯) | |
12 | ressnm.3 | . . . . 5 β’ 0 = (0gβπΊ) | |
13 | 1, 2, 12 | ress0g 18652 | . . . 4 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β 0 = (0gβπ»)) |
14 | 10, 11, 13 | oveq123d 7429 | . . 3 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β (π₯(distβπΊ) 0 ) = (π₯(distβπ»)(0gβπ»))) |
15 | 4, 14 | mpteq12dv 5239 | . 2 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β (π₯ β π΄ β¦ (π₯(distβπΊ) 0 )) = (π₯ β (Baseβπ») β¦ (π₯(distβπ»)(0gβπ»)))) |
16 | ressnm.4 | . . . . . 6 β’ π = (normβπΊ) | |
17 | 16, 2, 12, 7 | nmfval 24096 | . . . . 5 β’ π = (π₯ β π΅ β¦ (π₯(distβπΊ) 0 )) |
18 | 17 | reseq1i 5977 | . . . 4 β’ (π βΎ π΄) = ((π₯ β π΅ β¦ (π₯(distβπΊ) 0 )) βΎ π΄) |
19 | resmpt 6037 | . . . 4 β’ (π΄ β π΅ β ((π₯ β π΅ β¦ (π₯(distβπΊ) 0 )) βΎ π΄) = (π₯ β π΄ β¦ (π₯(distβπΊ) 0 ))) | |
20 | 18, 19 | eqtrid 2784 | . . 3 β’ (π΄ β π΅ β (π βΎ π΄) = (π₯ β π΄ β¦ (π₯(distβπΊ) 0 ))) |
21 | 20 | 3ad2ant3 1135 | . 2 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β (π βΎ π΄) = (π₯ β π΄ β¦ (π₯(distβπΊ) 0 ))) |
22 | eqid 2732 | . . . 4 β’ (normβπ») = (normβπ») | |
23 | eqid 2732 | . . . 4 β’ (Baseβπ») = (Baseβπ») | |
24 | eqid 2732 | . . . 4 β’ (0gβπ») = (0gβπ») | |
25 | eqid 2732 | . . . 4 β’ (distβπ») = (distβπ») | |
26 | 22, 23, 24, 25 | nmfval 24096 | . . 3 β’ (normβπ») = (π₯ β (Baseβπ») β¦ (π₯(distβπ»)(0gβπ»))) |
27 | 26 | a1i 11 | . 2 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β (normβπ») = (π₯ β (Baseβπ») β¦ (π₯(distβπ»)(0gβπ»)))) |
28 | 15, 21, 27 | 3eqtr4d 2782 | 1 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β (π βΎ π΄) = (normβπ»)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1087 = wceq 1541 β wcel 2106 Vcvv 3474 β wss 3948 β¦ cmpt 5231 βΎ cres 5678 βcfv 6543 (class class class)co 7408 Basecbs 17143 βΎs cress 17172 distcds 17205 0gc0g 17384 Mndcmnd 18624 normcnm 24084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-ds 17218 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-nm 24090 |
This theorem is referenced by: zringnm 32933 rezh 32946 |
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