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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 ℤ-module is normed. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
Ref | Expression |
---|---|
nmmulg.x | ⊢ 𝐵 = (Base‘𝑅) |
nmmulg.n | ⊢ 𝑁 = (norm‘𝑅) |
nmmulg.z | ⊢ 𝑍 = (ℤMod‘𝑅) |
nmmulg.t | ⊢ · = (.g‘𝑅) |
Ref | Expression |
---|---|
nmmulg | ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((abs‘𝑀) · (𝑁‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1136 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ ℤ) | |
2 | zringbas 21482 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
3 | nlmlmod 24715 | . . . . . . . . 9 ⊢ (𝑍 ∈ NrmMod → 𝑍 ∈ LMod) | |
4 | nmmulg.z | . . . . . . . . . 10 ⊢ 𝑍 = (ℤMod‘𝑅) | |
5 | 4 | zlmlmod 21555 | . . . . . . . . 9 ⊢ (𝑅 ∈ Abel ↔ 𝑍 ∈ LMod) |
6 | 3, 5 | sylibr 234 | . . . . . . . 8 ⊢ (𝑍 ∈ NrmMod → 𝑅 ∈ Abel) |
7 | 6 | 3ad2ant1 1132 | . . . . . . 7 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Abel) |
8 | 4 | zlmsca 21553 | . . . . . . 7 ⊢ (𝑅 ∈ Abel → ℤring = (Scalar‘𝑍)) |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ℤring = (Scalar‘𝑍)) |
10 | 9 | fveq2d 6911 | . . . . 5 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (Base‘ℤring) = (Base‘(Scalar‘𝑍))) |
11 | 2, 10 | eqtrid 2787 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ℤ = (Base‘(Scalar‘𝑍))) |
12 | 1, 11 | eleqtrd 2841 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ (Base‘(Scalar‘𝑍))) |
13 | nmmulg.x | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
14 | 4, 13 | zlmbas 21547 | . . . 4 ⊢ 𝐵 = (Base‘𝑍) |
15 | eqid 2735 | . . . 4 ⊢ (norm‘𝑍) = (norm‘𝑍) | |
16 | nmmulg.t | . . . . 5 ⊢ · = (.g‘𝑅) | |
17 | 4, 16 | zlmvsca 21554 | . . . 4 ⊢ · = ( ·𝑠 ‘𝑍) |
18 | eqid 2735 | . . . 4 ⊢ (Scalar‘𝑍) = (Scalar‘𝑍) | |
19 | eqid 2735 | . . . 4 ⊢ (Base‘(Scalar‘𝑍)) = (Base‘(Scalar‘𝑍)) | |
20 | eqid 2735 | . . . 4 ⊢ (norm‘(Scalar‘𝑍)) = (norm‘(Scalar‘𝑍)) | |
21 | 14, 15, 17, 18, 19, 20 | nmvs 24713 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ (Base‘(Scalar‘𝑍)) ∧ 𝑋 ∈ 𝐵) → ((norm‘𝑍)‘(𝑀 · 𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋))) |
22 | 12, 21 | syld3an2 1410 | . 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((norm‘𝑍)‘(𝑀 · 𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋))) |
23 | nmmulg.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑅) | |
24 | 4, 23 | zlmnm 33927 | . . . 4 ⊢ (𝑅 ∈ Abel → 𝑁 = (norm‘𝑍)) |
25 | 7, 24 | syl 17 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑁 = (norm‘𝑍)) |
26 | 25 | fveq1d 6909 | . 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((norm‘𝑍)‘(𝑀 · 𝑋))) |
27 | zzsnm 33920 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) = ((norm‘ℤring)‘𝑀)) | |
28 | 27 | 3ad2ant2 1133 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (abs‘𝑀) = ((norm‘ℤring)‘𝑀)) |
29 | 9 | fveq2d 6911 | . . . . 5 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (norm‘ℤring) = (norm‘(Scalar‘𝑍))) |
30 | 29 | fveq1d 6909 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((norm‘ℤring)‘𝑀) = ((norm‘(Scalar‘𝑍))‘𝑀)) |
31 | 28, 30 | eqtrd 2775 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (abs‘𝑀) = ((norm‘(Scalar‘𝑍))‘𝑀)) |
32 | 25 | fveq1d 6909 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ((norm‘𝑍)‘𝑋)) |
33 | 31, 32 | oveq12d 7449 | . 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((abs‘𝑀) · (𝑁‘𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋))) |
34 | 22, 26, 33 | 3eqtr4d 2785 | 1 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((abs‘𝑀) · (𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 · cmul 11158 ℤcz 12611 abscabs 15270 Basecbs 17245 Scalarcsca 17301 .gcmg 19098 Abelcabl 19814 LModclmod 20875 ℤringczring 21475 ℤModczlm 21529 normcnm 24605 NrmModcnlm 24609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-mulg 19099 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-cnfld 21383 df-zring 21476 df-zlm 21533 df-nm 24611 df-nlm 24615 |
This theorem is referenced by: zrhnm 33930 |
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