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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 17183 | . . . 4 β’ (π΄ β π΅ β π΄ = (Baseβπ»)) |
4 | 3 | 3ad2ant3 1132 | . . 3 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β π΄ = (Baseβπ»)) |
5 | 2 | fvexi 6896 | . . . . . . 7 β’ π΅ β V |
6 | 5 | ssex 5312 | . . . . . 6 β’ (π΄ β π΅ β π΄ β V) |
7 | eqid 2724 | . . . . . . 7 β’ (distβπΊ) = (distβπΊ) | |
8 | 1, 7 | ressds 17356 | . . . . . 6 β’ (π΄ β V β (distβπΊ) = (distβπ»)) |
9 | 6, 8 | syl 17 | . . . . 5 β’ (π΄ β π΅ β (distβπΊ) = (distβπ»)) |
10 | 9 | 3ad2ant3 1132 | . . . 4 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β (distβπΊ) = (distβπ»)) |
11 | eqidd 2725 | . . . 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 18687 | . . . 4 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β 0 = (0gβπ»)) |
14 | 10, 11, 13 | oveq123d 7423 | . . 3 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β (π₯(distβπΊ) 0 ) = (π₯(distβπ»)(0gβπ»))) |
15 | 4, 14 | mpteq12dv 5230 | . 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 24421 | . . . . 5 β’ π = (π₯ β π΅ β¦ (π₯(distβπΊ) 0 )) |
18 | 17 | reseq1i 5968 | . . . 4 β’ (π βΎ π΄) = ((π₯ β π΅ β¦ (π₯(distβπΊ) 0 )) βΎ π΄) |
19 | resmpt 6028 | . . . 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 2776 | . . 3 β’ (π΄ β π΅ β (π βΎ π΄) = (π₯ β π΄ β¦ (π₯(distβπΊ) 0 ))) |
21 | 20 | 3ad2ant3 1132 | . 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 2724 | . . . 4 β’ (normβπ») = (normβπ») | |
23 | eqid 2724 | . . . 4 β’ (Baseβπ») = (Baseβπ») | |
24 | eqid 2724 | . . . 4 β’ (0gβπ») = (0gβπ») | |
25 | eqid 2724 | . . . 4 β’ (distβπ») = (distβπ») | |
26 | 22, 23, 24, 25 | nmfval 24421 | . . 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 2774 | 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 1084 = wceq 1533 β wcel 2098 Vcvv 3466 β wss 3941 β¦ cmpt 5222 βΎ cres 5669 βcfv 6534 (class class class)co 7402 Basecbs 17145 βΎs cress 17174 distcds 17207 0gc0g 17386 Mndcmnd 18659 normcnm 24409 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-ds 17220 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-nm 24415 |
This theorem is referenced by: zringnm 33430 rezh 33443 |
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