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 1189 | . . . . 5 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑅 ∈ NzRing) | |
2 | nzrring 20036 | . . . . 5 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑅 ∈ Ring) |
4 | simpr 487 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ) | |
5 | zrhnm.1 | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
6 | eqid 2823 | . . . . . 6 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
7 | eqid 2823 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
8 | 5, 6, 7 | zrhmulg 20659 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ) → (𝐿‘𝑀) = (𝑀(.g‘𝑅)(1r‘𝑅))) |
9 | 8 | fveq2d 6676 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅)))) |
10 | 3, 4, 9 | syl2anc 586 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅)))) |
11 | simpl1 1187 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ NrmMod) | |
12 | nmmulg.x | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
13 | 12, 7 | ringidcl 19320 | . . . . 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 31211 | . . . 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 1367 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅))) = ((abs‘𝑀) · (𝑁‘(1r‘𝑅)))) |
19 | 16, 15 | zlmnm 31209 | . . . . . . 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 6674 | . . . . 5 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(1r‘𝑅)) = ((norm‘𝑍)‘(1r‘𝑅))) |
22 | simpl2 1188 | . . . . . 6 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ NrmRing) | |
23 | nrgring 23274 | . . . . . . . 8 ⊢ (𝑍 ∈ NrmRing → 𝑍 ∈ Ring) | |
24 | 22, 23 | syl 17 | . . . . . . 7 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ Ring) |
25 | eqid 2823 | . . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
26 | 7, 25 | nzrnz 20035 | . . . . . . . 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 31206 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑍) |
29 | 16, 25 | zlm0 31205 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑍) |
30 | 28, 29 | isnzr 20034 | . . . . . . 7 ⊢ (𝑍 ∈ NzRing ↔ (𝑍 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
31 | 24, 27, 30 | sylanbrc 585 | . . . . . 6 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ NzRing) |
32 | eqid 2823 | . . . . . . 7 ⊢ (norm‘𝑍) = (norm‘𝑍) | |
33 | 32, 28 | nm1 23278 | . . . . . 6 ⊢ ((𝑍 ∈ NrmRing ∧ 𝑍 ∈ NzRing) → ((norm‘𝑍)‘(1r‘𝑅)) = 1) |
34 | 22, 31, 33 | syl2anc 586 | . . . . 5 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → ((norm‘𝑍)‘(1r‘𝑅)) = 1) |
35 | 21, 34 | eqtrd 2858 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(1r‘𝑅)) = 1) |
36 | 35 | oveq2d 7174 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → ((abs‘𝑀) · (𝑁‘(1r‘𝑅))) = ((abs‘𝑀) · 1)) |
37 | 10, 18, 36 | 3eqtrd 2862 | . 2 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = ((abs‘𝑀) · 1)) |
38 | 4 | zcnd 12091 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℂ) |
39 | abscl 14640 | . . . 4 ⊢ (𝑀 ∈ ℂ → (abs‘𝑀) ∈ ℝ) | |
40 | 39 | recnd 10671 | . . 3 ⊢ (𝑀 ∈ ℂ → (abs‘𝑀) ∈ ℂ) |
41 | mulid1 10641 | . . 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 2858 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 1c1 10540 · cmul 10544 ℤcz 11984 abscabs 14595 Basecbs 16485 0gc0g 16715 .gcmg 18226 1rcur 19253 Ringcrg 19299 NzRingcnzr 20032 ℤRHomczrh 20649 ℤModczlm 20650 normcnm 23188 NrmRingcnrg 23191 NrmModcnlm 23192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-ico 12747 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-minusg 18109 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-rnghom 19469 df-subrg 19535 df-abv 19590 df-lmod 19638 df-nzr 20033 df-cnfld 20548 df-zring 20620 df-zrh 20653 df-zlm 20654 df-nm 23194 df-nrg 23197 df-nlm 23198 |
This theorem is referenced by: qqhnm 31233 |
Copyright terms: Public domain | W3C validator |