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Mirrors > Home > MPE Home > Th. List > nmdvr | Structured version Visualization version GIF version |
Description: The norm of a division in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
nmdvr.x | ⊢ 𝑋 = (Base‘𝑅) |
nmdvr.n | ⊢ 𝑁 = (norm‘𝑅) |
nmdvr.u | ⊢ 𝑈 = (Unit‘𝑅) |
nmdvr.d | ⊢ / = (/r‘𝑅) |
Ref | Expression |
---|---|
nmdvr | ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴 / 𝐵)) = ((𝑁‘𝐴) / (𝑁‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 790 | . . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NrmRing) | |
2 | simprl 794 | . . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐴 ∈ 𝑋) | |
3 | nrgring 22467 | . . . . . 6 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring) | |
4 | 3 | ad2antrr 762 | . . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ Ring) |
5 | simprr 796 | . . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐵 ∈ 𝑈) | |
6 | nmdvr.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
7 | eqid 2622 | . . . . . 6 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
8 | nmdvr.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝑅) | |
9 | 6, 7, 8 | ringinvcl 18676 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐵 ∈ 𝑈) → ((invr‘𝑅)‘𝐵) ∈ 𝑋) |
10 | 4, 5, 9 | syl2anc 693 | . . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → ((invr‘𝑅)‘𝐵) ∈ 𝑋) |
11 | nmdvr.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑅) | |
12 | eqid 2622 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
13 | 8, 11, 12 | nmmul 22468 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝐴 ∈ 𝑋 ∧ ((invr‘𝑅)‘𝐵) ∈ 𝑋) → (𝑁‘(𝐴(.r‘𝑅)((invr‘𝑅)‘𝐵))) = ((𝑁‘𝐴) · (𝑁‘((invr‘𝑅)‘𝐵)))) |
14 | 1, 2, 10, 13 | syl3anc 1326 | . . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴(.r‘𝑅)((invr‘𝑅)‘𝐵))) = ((𝑁‘𝐴) · (𝑁‘((invr‘𝑅)‘𝐵)))) |
15 | simplr 792 | . . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NzRing) | |
16 | 11, 6, 7 | nminvr 22473 | . . . . 5 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐵 ∈ 𝑈) → (𝑁‘((invr‘𝑅)‘𝐵)) = (1 / (𝑁‘𝐵))) |
17 | 1, 15, 5, 16 | syl3anc 1326 | . . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘((invr‘𝑅)‘𝐵)) = (1 / (𝑁‘𝐵))) |
18 | 17 | oveq2d 6666 | . . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → ((𝑁‘𝐴) · (𝑁‘((invr‘𝑅)‘𝐵))) = ((𝑁‘𝐴) · (1 / (𝑁‘𝐵)))) |
19 | 14, 18 | eqtrd 2656 | . 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴(.r‘𝑅)((invr‘𝑅)‘𝐵))) = ((𝑁‘𝐴) · (1 / (𝑁‘𝐵)))) |
20 | nmdvr.d | . . . . 5 ⊢ / = (/r‘𝑅) | |
21 | 8, 12, 6, 7, 20 | dvrval 18685 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈) → (𝐴 / 𝐵) = (𝐴(.r‘𝑅)((invr‘𝑅)‘𝐵))) |
22 | 21 | adantl 482 | . . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝐴 / 𝐵) = (𝐴(.r‘𝑅)((invr‘𝑅)‘𝐵))) |
23 | 22 | fveq2d 6195 | . 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴 / 𝐵)) = (𝑁‘(𝐴(.r‘𝑅)((invr‘𝑅)‘𝐵)))) |
24 | nrgngp 22466 | . . . . . 6 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp) | |
25 | 24 | ad2antrr 762 | . . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝑅 ∈ NrmGrp) |
26 | 8, 11 | nmcl 22420 | . . . . 5 ⊢ ((𝑅 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
27 | 25, 2, 26 | syl2anc 693 | . . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐴) ∈ ℝ) |
28 | 27 | recnd 10068 | . . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐴) ∈ ℂ) |
29 | 8, 6 | unitss 18660 | . . . . . 6 ⊢ 𝑈 ⊆ 𝑋 |
30 | 29, 5 | sseldi 3601 | . . . . 5 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → 𝐵 ∈ 𝑋) |
31 | 8, 11 | nmcl 22420 | . . . . 5 ⊢ ((𝑅 ∈ NrmGrp ∧ 𝐵 ∈ 𝑋) → (𝑁‘𝐵) ∈ ℝ) |
32 | 25, 30, 31 | syl2anc 693 | . . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ∈ ℝ) |
33 | 32 | recnd 10068 | . . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ∈ ℂ) |
34 | 11, 6 | unitnmn0 22472 | . . . . 5 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐵 ∈ 𝑈) → (𝑁‘𝐵) ≠ 0) |
35 | 34 | 3expa 1265 | . . . 4 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝐵 ∈ 𝑈) → (𝑁‘𝐵) ≠ 0) |
36 | 35 | adantrl 752 | . . 3 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘𝐵) ≠ 0) |
37 | 28, 33, 36 | divrecd 10804 | . 2 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → ((𝑁‘𝐴) / (𝑁‘𝐵)) = ((𝑁‘𝐴) · (1 / (𝑁‘𝐵)))) |
38 | 19, 23, 37 | 3eqtr4d 2666 | 1 ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴 / 𝐵)) = ((𝑁‘𝐴) / (𝑁‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 / cdiv 10684 Basecbs 15857 .rcmulr 15942 Ringcrg 18547 Unitcui 18639 invrcinvr 18671 /rcdvr 18682 NzRingcnzr 19257 normcnm 22381 NrmGrpcngp 22382 NrmRingcnrg 22384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ico 12181 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-topgen 16104 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-abv 18817 df-nzr 19258 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-xms 22125 df-ms 22126 df-nm 22387 df-ngp 22388 df-nrg 22390 |
This theorem is referenced by: qqhnm 30034 |
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