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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zrhnm | Structured version Visualization version GIF version |
Description: The norm of the image by ℤRHom of an integer in a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
Ref | Expression |
---|---|
nmmulg.x | ⊢ 𝐵 = (Base‘𝑅) |
nmmulg.n | ⊢ 𝑁 = (norm‘𝑅) |
nmmulg.z | ⊢ 𝑍 = (ℤMod‘𝑅) |
zrhnm.1 | ⊢ 𝐿 = (ℤRHom‘𝑅) |
Ref | Expression |
---|---|
zrhnm | ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (abs‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl3 1192 | . . . . 5 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑅 ∈ NzRing) | |
2 | nzrring 20533 | . . . . 5 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑅 ∈ Ring) |
4 | simpr 484 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ) | |
5 | zrhnm.1 | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
6 | eqid 2735 | . . . . . 6 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
7 | eqid 2735 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
8 | 5, 6, 7 | zrhmulg 21538 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ) → (𝐿‘𝑀) = (𝑀(.g‘𝑅)(1r‘𝑅))) |
9 | 8 | fveq2d 6911 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅)))) |
10 | 3, 4, 9 | syl2anc 584 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅)))) |
11 | simpl1 1190 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ NrmMod) | |
12 | nmmulg.x | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
13 | 12, 7 | ringidcl 20280 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
14 | 3, 13 | syl 17 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (1r‘𝑅) ∈ 𝐵) |
15 | nmmulg.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑅) | |
16 | nmmulg.z | . . . . 5 ⊢ 𝑍 = (ℤMod‘𝑅) | |
17 | 12, 15, 16, 6 | nmmulg 33929 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ (1r‘𝑅) ∈ 𝐵) → (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅))) = ((abs‘𝑀) · (𝑁‘(1r‘𝑅)))) |
18 | 11, 4, 14, 17 | syl3anc 1370 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅))) = ((abs‘𝑀) · (𝑁‘(1r‘𝑅)))) |
19 | 16, 15 | zlmnm 33927 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → 𝑁 = (norm‘𝑍)) |
20 | 1, 19 | syl 17 | . . . . . 6 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑁 = (norm‘𝑍)) |
21 | 20 | fveq1d 6909 | . . . . 5 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(1r‘𝑅)) = ((norm‘𝑍)‘(1r‘𝑅))) |
22 | simpl2 1191 | . . . . . 6 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ NrmRing) | |
23 | nrgring 24700 | . . . . . . . 8 ⊢ (𝑍 ∈ NrmRing → 𝑍 ∈ Ring) | |
24 | 22, 23 | syl 17 | . . . . . . 7 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ Ring) |
25 | eqid 2735 | . . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
26 | 7, 25 | nzrnz 20532 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
27 | 1, 26 | syl 17 | . . . . . . 7 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (1r‘𝑅) ≠ (0g‘𝑅)) |
28 | 16, 7 | zlm1 33922 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑍) |
29 | 16, 25 | zlm0 33921 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑍) |
30 | 28, 29 | isnzr 20531 | . . . . . . 7 ⊢ (𝑍 ∈ NzRing ↔ (𝑍 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
31 | 24, 27, 30 | sylanbrc 583 | . . . . . 6 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ NzRing) |
32 | eqid 2735 | . . . . . . 7 ⊢ (norm‘𝑍) = (norm‘𝑍) | |
33 | 32, 28 | nm1 24704 | . . . . . 6 ⊢ ((𝑍 ∈ NrmRing ∧ 𝑍 ∈ NzRing) → ((norm‘𝑍)‘(1r‘𝑅)) = 1) |
34 | 22, 31, 33 | syl2anc 584 | . . . . 5 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → ((norm‘𝑍)‘(1r‘𝑅)) = 1) |
35 | 21, 34 | eqtrd 2775 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(1r‘𝑅)) = 1) |
36 | 35 | oveq2d 7447 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → ((abs‘𝑀) · (𝑁‘(1r‘𝑅))) = ((abs‘𝑀) · 1)) |
37 | 10, 18, 36 | 3eqtrd 2779 | . 2 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = ((abs‘𝑀) · 1)) |
38 | 4 | zcnd 12721 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℂ) |
39 | abscl 15314 | . . . 4 ⊢ (𝑀 ∈ ℂ → (abs‘𝑀) ∈ ℝ) | |
40 | 39 | recnd 11287 | . . 3 ⊢ (𝑀 ∈ ℂ → (abs‘𝑀) ∈ ℂ) |
41 | mulrid 11257 | . . 3 ⊢ ((abs‘𝑀) ∈ ℂ → ((abs‘𝑀) · 1) = (abs‘𝑀)) | |
42 | 38, 40, 41 | 3syl 18 | . 2 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → ((abs‘𝑀) · 1) = (abs‘𝑀)) |
43 | 37, 42 | eqtrd 2775 | 1 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (abs‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 1c1 11154 · cmul 11158 ℤcz 12611 abscabs 15270 Basecbs 17245 0gc0g 17486 .gcmg 19098 1rcur 20199 Ringcrg 20251 NzRingcnzr 20529 ℤRHomczrh 21528 ℤModczlm 21529 normcnm 24605 NrmRingcnrg 24608 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-rep 5285 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-map 8867 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-ico 13390 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-mhm 18809 df-grp 18967 df-minusg 18968 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-rhm 20489 df-nzr 20530 df-subrng 20563 df-subrg 20587 df-abv 20827 df-lmod 20877 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-zlm 21533 df-nm 24611 df-nrg 24614 df-nlm 24615 |
This theorem is referenced by: qqhnm 33953 |
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