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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 17212 | . . . 4 β’ (π΄ β π΅ β π΄ = (Baseβπ»)) |
4 | 3 | 3ad2ant3 1133 | . . 3 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β π΄ = (Baseβπ»)) |
5 | 2 | fvexi 6906 | . . . . . . 7 β’ π΅ β V |
6 | 5 | ssex 5316 | . . . . . 6 β’ (π΄ β π΅ β π΄ β V) |
7 | eqid 2728 | . . . . . . 7 β’ (distβπΊ) = (distβπΊ) | |
8 | 1, 7 | ressds 17385 | . . . . . 6 β’ (π΄ β V β (distβπΊ) = (distβπ»)) |
9 | 6, 8 | syl 17 | . . . . 5 β’ (π΄ β π΅ β (distβπΊ) = (distβπ»)) |
10 | 9 | 3ad2ant3 1133 | . . . 4 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β (distβπΊ) = (distβπ»)) |
11 | eqidd 2729 | . . . 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 18716 | . . . 4 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β 0 = (0gβπ»)) |
14 | 10, 11, 13 | oveq123d 7436 | . . 3 β’ ((πΊ β Mnd β§ 0 β π΄ β§ π΄ β π΅) β (π₯(distβπΊ) 0 ) = (π₯(distβπ»)(0gβπ»))) |
15 | 4, 14 | mpteq12dv 5234 | . 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 24491 | . . . . 5 β’ π = (π₯ β π΅ β¦ (π₯(distβπΊ) 0 )) |
18 | 17 | reseq1i 5976 | . . . 4 β’ (π βΎ π΄) = ((π₯ β π΅ β¦ (π₯(distβπΊ) 0 )) βΎ π΄) |
19 | resmpt 6036 | . . . 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 2780 | . . 3 β’ (π΄ β π΅ β (π βΎ π΄) = (π₯ β π΄ β¦ (π₯(distβπΊ) 0 ))) |
21 | 20 | 3ad2ant3 1133 | . 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 2728 | . . . 4 β’ (normβπ») = (normβπ») | |
23 | eqid 2728 | . . . 4 β’ (Baseβπ») = (Baseβπ») | |
24 | eqid 2728 | . . . 4 β’ (0gβπ») = (0gβπ») | |
25 | eqid 2728 | . . . 4 β’ (distβπ») = (distβπ») | |
26 | 22, 23, 24, 25 | nmfval 24491 | . . 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 2778 | 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 1085 = wceq 1534 β wcel 2099 Vcvv 3470 β wss 3945 β¦ cmpt 5226 βΎ cres 5675 βcfv 6543 (class class class)co 7415 Basecbs 17174 βΎs cress 17203 distcds 17236 0gc0g 17415 Mndcmnd 18688 normcnm 24479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-ds 17249 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-nm 24485 |
This theorem is referenced by: zringnm 33554 rezh 33567 |
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