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Mathbox for Thierry Arnoux |
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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 1135 | . . . 4 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β π β β€) | |
2 | zringbas 21379 | . . . . 5 β’ β€ = (Baseββ€ring) | |
3 | nlmlmod 24608 | . . . . . . . . 9 β’ (π β NrmMod β π β LMod) | |
4 | nmmulg.z | . . . . . . . . . 10 β’ π = (β€Modβπ ) | |
5 | 4 | zlmlmod 21452 | . . . . . . . . 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 1131 | . . . . . . 7 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β π β Abel) |
8 | 4 | zlmsca 21450 | . . . . . . 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 6901 | . . . . 5 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (Baseββ€ring) = (Baseβ(Scalarβπ))) |
11 | 2, 10 | eqtrid 2780 | . . . 4 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β β€ = (Baseβ(Scalarβπ))) |
12 | 1, 11 | eleqtrd 2831 | . . 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 21444 | . . . 4 β’ π΅ = (Baseβπ) |
15 | eqid 2728 | . . . 4 β’ (normβπ) = (normβπ) | |
16 | nmmulg.t | . . . . 5 β’ Β· = (.gβπ ) | |
17 | 4, 16 | zlmvsca 21451 | . . . 4 β’ Β· = ( Β·π βπ) |
18 | eqid 2728 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
19 | eqid 2728 | . . . 4 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
20 | eqid 2728 | . . . 4 β’ (normβ(Scalarβπ)) = (normβ(Scalarβπ)) | |
21 | 14, 15, 17, 18, 19, 20 | nmvs 24606 | . . 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 1409 | . 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 33567 | . . . 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 6899 | . 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 33560 | . . . . 5 β’ (π β β€ β (absβπ) = ((normββ€ring)βπ)) | |
28 | 27 | 3ad2ant2 1132 | . . . 4 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (absβπ) = ((normββ€ring)βπ)) |
29 | 9 | fveq2d 6901 | . . . . 5 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (normββ€ring) = (normβ(Scalarβπ))) |
30 | 29 | fveq1d 6899 | . . . 4 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β ((normββ€ring)βπ) = ((normβ(Scalarβπ))βπ)) |
31 | 28, 30 | eqtrd 2768 | . . 3 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (absβπ) = ((normβ(Scalarβπ))βπ)) |
32 | 25 | fveq1d 6899 | . . 3 β’ ((π β NrmMod β§ π β β€ β§ π β π΅) β (πβπ) = ((normβπ)βπ)) |
33 | 31, 32 | oveq12d 7438 | . 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 2778 | 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 1085 = wceq 1534 β wcel 2099 βcfv 6548 (class class class)co 7420 Β· cmul 11144 β€cz 12589 abscabs 15214 Basecbs 17180 Scalarcsca 17236 .gcmg 19023 Abelcabl 19736 LModclmod 20743 β€ringczring 21372 β€Modczlm 21426 normcnm 24498 NrmModcnlm 24502 |
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 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218 ax-mulf 11219 |
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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-rp 13008 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-mulg 19024 df-subg 19078 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-subrng 20483 df-subrg 20508 df-lmod 20745 df-cnfld 21280 df-zring 21373 df-zlm 21430 df-nm 24504 df-nlm 24508 |
This theorem is referenced by: zrhnm 33570 |
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