| Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
| 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 1194 | . . . . 5 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑅 ∈ NzRing) | |
| 2 | nzrring 20516 | . . . . 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 2737 | . . . . . 6 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
| 7 | eqid 2737 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 8 | 5, 6, 7 | zrhmulg 21520 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ) → (𝐿‘𝑀) = (𝑀(.g‘𝑅)(1r‘𝑅))) |
| 9 | 8 | fveq2d 6910 | . . . 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 1192 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ NrmMod) | |
| 12 | nmmulg.x | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
| 13 | 12, 7 | ringidcl 20262 | . . . . 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 33967 | . . . 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 1373 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅))) = ((abs‘𝑀) · (𝑁‘(1r‘𝑅)))) |
| 19 | 16, 15 | zlmnm 33965 | . . . . . . 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 6908 | . . . . 5 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(1r‘𝑅)) = ((norm‘𝑍)‘(1r‘𝑅))) |
| 22 | simpl2 1193 | . . . . . 6 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ NrmRing) | |
| 23 | nrgring 24684 | . . . . . . . 8 ⊢ (𝑍 ∈ NrmRing → 𝑍 ∈ Ring) | |
| 24 | 22, 23 | syl 17 | . . . . . . 7 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ Ring) |
| 25 | eqid 2737 | . . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 26 | 7, 25 | nzrnz 20515 | . . . . . . . 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 33960 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑍) |
| 29 | 16, 25 | zlm0 33959 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑍) |
| 30 | 28, 29 | isnzr 20514 | . . . . . . 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 2737 | . . . . . . 7 ⊢ (norm‘𝑍) = (norm‘𝑍) | |
| 33 | 32, 28 | nm1 24688 | . . . . . 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 2777 | . . . 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 2781 | . 2 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = ((abs‘𝑀) · 1)) |
| 38 | 4 | zcnd 12723 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℂ) |
| 39 | abscl 15317 | . . . 4 ⊢ (𝑀 ∈ ℂ → (abs‘𝑀) ∈ ℝ) | |
| 40 | 39 | recnd 11289 | . . 3 ⊢ (𝑀 ∈ ℂ → (abs‘𝑀) ∈ ℂ) |
| 41 | mulrid 11259 | . . 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 2777 | 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 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 1c1 11156 · cmul 11160 ℤcz 12613 abscabs 15273 Basecbs 17247 0gc0g 17484 .gcmg 19085 1rcur 20178 Ringcrg 20230 NzRingcnzr 20512 ℤRHomczrh 21510 ℤModczlm 21511 normcnm 24589 NrmRingcnrg 24592 NrmModcnlm 24593 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-ico 13393 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-rhm 20472 df-nzr 20513 df-subrng 20546 df-subrg 20570 df-abv 20810 df-lmod 20860 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-zlm 21515 df-nm 24595 df-nrg 24598 df-nlm 24599 |
| This theorem is referenced by: qqhnm 33991 |
| Copyright terms: Public domain | W3C validator |