![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > nmmulg | Structured version Visualization version GIF version |
Description: The norm of a group product, provided the β€-module is normed. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
Ref | Expression |
---|---|
nmmulg.x | β’ π΅ = (Baseβπ ) |
nmmulg.n | β’ π = (normβπ ) |
nmmulg.z | β’ π = (β€Modβπ ) |
nmmulg.t | β’ Β· = (.gβπ ) |
Ref | Expression |
---|---|
nmmulg | β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (πβ(π Β· π)) = ((absβπ) Β· (πβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1134 | . . . 4 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β π β β€) | |
2 | zringbas 21336 | . . . . 5 β’ β€ = (Baseββ€ring) | |
3 | nlmlmod 24546 | . . . . . . . . 9 β’ (π β NrmMod β π β LMod) | |
4 | nmmulg.z | . . . . . . . . . 10 β’ π = (β€Modβπ ) | |
5 | 4 | zlmlmod 21409 | . . . . . . . . 9 β’ (π β Abel β π β LMod) |
6 | 3, 5 | sylibr 233 | . . . . . . . 8 β’ (π β NrmMod β π β Abel) |
7 | 6 | 3ad2ant1 1130 | . . . . . . 7 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β π β Abel) |
8 | 4 | zlmsca 21407 | . . . . . . 7 β’ (π β Abel β β€ring = (Scalarβπ)) |
9 | 7, 8 | syl 17 | . . . . . 6 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β β€ring = (Scalarβπ)) |
10 | 9 | fveq2d 6888 | . . . . 5 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (Baseββ€ring) = (Baseβ(Scalarβπ))) |
11 | 2, 10 | eqtrid 2778 | . . . 4 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β β€ = (Baseβ(Scalarβπ))) |
12 | 1, 11 | eleqtrd 2829 | . . 3 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β π β (Baseβ(Scalarβπ))) |
13 | nmmulg.x | . . . . 5 β’ π΅ = (Baseβπ ) | |
14 | 4, 13 | zlmbas 21401 | . . . 4 β’ π΅ = (Baseβπ) |
15 | eqid 2726 | . . . 4 β’ (normβπ) = (normβπ) | |
16 | nmmulg.t | . . . . 5 β’ Β· = (.gβπ ) | |
17 | 4, 16 | zlmvsca 21408 | . . . 4 β’ Β· = ( Β·π βπ) |
18 | eqid 2726 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
19 | eqid 2726 | . . . 4 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
20 | eqid 2726 | . . . 4 β’ (normβ(Scalarβπ)) = (normβ(Scalarβπ)) | |
21 | 14, 15, 17, 18, 19, 20 | nmvs 24544 | . . 3 β’ ((π β NrmMod β§ π β (Baseβ(Scalarβπ)) β§ π β π΅) β ((normβπ)β(π Β· π)) = (((normβ(Scalarβπ))βπ) Β· ((normβπ)βπ))) |
22 | 12, 21 | syld3an2 1408 | . 2 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β ((normβπ)β(π Β· π)) = (((normβ(Scalarβπ))βπ) Β· ((normβπ)βπ))) |
23 | nmmulg.n | . . . . 5 β’ π = (normβπ ) | |
24 | 4, 23 | zlmnm 33476 | . . . 4 β’ (π β Abel β π = (normβπ)) |
25 | 7, 24 | syl 17 | . . 3 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β π = (normβπ)) |
26 | 25 | fveq1d 6886 | . 2 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (πβ(π Β· π)) = ((normβπ)β(π Β· π))) |
27 | zzsnm 33469 | . . . . 5 β’ (π β β€ β (absβπ) = ((normββ€ring)βπ)) | |
28 | 27 | 3ad2ant2 1131 | . . . 4 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (absβπ) = ((normββ€ring)βπ)) |
29 | 9 | fveq2d 6888 | . . . . 5 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (normββ€ring) = (normβ(Scalarβπ))) |
30 | 29 | fveq1d 6886 | . . . 4 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β ((normββ€ring)βπ) = ((normβ(Scalarβπ))βπ)) |
31 | 28, 30 | eqtrd 2766 | . . 3 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (absβπ) = ((normβ(Scalarβπ))βπ)) |
32 | 25 | fveq1d 6886 | . . 3 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (πβπ) = ((normβπ)βπ)) |
33 | 31, 32 | oveq12d 7422 | . 2 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β ((absβπ) Β· (πβπ)) = (((normβ(Scalarβπ))βπ) Β· ((normβπ)βπ))) |
34 | 22, 26, 33 | 3eqtr4d 2776 | 1 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (πβ(π Β· π)) = ((absβπ) Β· (πβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 Β· cmul 11114 β€cz 12559 abscabs 15185 Basecbs 17151 Scalarcsca 17207 .gcmg 18993 Abelcabl 19699 LModclmod 20704 β€ringczring 21329 β€Modczlm 21383 normcnm 24436 NrmModcnlm 24440 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-rp 12978 df-fz 13488 df-fzo 13631 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-mulg 18994 df-subg 19048 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-subrng 20444 df-subrg 20469 df-lmod 20706 df-cnfld 21237 df-zring 21330 df-zlm 21387 df-nm 24442 df-nlm 24446 |
This theorem is referenced by: zrhnm 33479 |
Copyright terms: Public domain | W3C validator |