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Mirrors > Home > MPE Home > Th. List > nminvr | Structured version Visualization version GIF version |
Description: The norm of an inverse in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
nminvr.n | ⊢ 𝑁 = (norm‘𝑅) |
nminvr.u | ⊢ 𝑈 = (Unit‘𝑅) |
nminvr.i | ⊢ 𝐼 = (invr‘𝑅) |
Ref | Expression |
---|---|
nminvr | ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘(𝐼‘𝐴)) = (1 / (𝑁‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrgngp 23273 | . . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp) | |
2 | 1 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → 𝑅 ∈ NrmGrp) |
3 | eqid 2823 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | nminvr.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
5 | 3, 4 | unitcl 19411 | . . . . 5 ⊢ (𝐴 ∈ 𝑈 → 𝐴 ∈ (Base‘𝑅)) |
6 | 5 | 3ad2ant3 1131 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ (Base‘𝑅)) |
7 | nminvr.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑅) | |
8 | 3, 7 | nmcl 23227 | . . . 4 ⊢ ((𝑅 ∈ NrmGrp ∧ 𝐴 ∈ (Base‘𝑅)) → (𝑁‘𝐴) ∈ ℝ) |
9 | 2, 6, 8 | syl2anc 586 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘𝐴) ∈ ℝ) |
10 | 9 | recnd 10671 | . 2 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘𝐴) ∈ ℂ) |
11 | nzrring 20036 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
12 | 11 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → 𝑅 ∈ Ring) |
13 | simp3 1134 | . . . . 5 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝑈) | |
14 | nminvr.i | . . . . . 6 ⊢ 𝐼 = (invr‘𝑅) | |
15 | 4, 14, 3 | ringinvcl 19428 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝑈) → (𝐼‘𝐴) ∈ (Base‘𝑅)) |
16 | 12, 13, 15 | syl2anc 586 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝐼‘𝐴) ∈ (Base‘𝑅)) |
17 | 3, 7 | nmcl 23227 | . . . 4 ⊢ ((𝑅 ∈ NrmGrp ∧ (𝐼‘𝐴) ∈ (Base‘𝑅)) → (𝑁‘(𝐼‘𝐴)) ∈ ℝ) |
18 | 2, 16, 17 | syl2anc 586 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘(𝐼‘𝐴)) ∈ ℝ) |
19 | 18 | recnd 10671 | . 2 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘(𝐼‘𝐴)) ∈ ℂ) |
20 | 7, 4 | unitnmn0 23279 | . 2 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘𝐴) ≠ 0) |
21 | eqid 2823 | . . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
22 | eqid 2823 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
23 | 4, 14, 21, 22 | unitrinv 19430 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝑈) → (𝐴(.r‘𝑅)(𝐼‘𝐴)) = (1r‘𝑅)) |
24 | 12, 13, 23 | syl2anc 586 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝐴(.r‘𝑅)(𝐼‘𝐴)) = (1r‘𝑅)) |
25 | 24 | fveq2d 6676 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘(𝐴(.r‘𝑅)(𝐼‘𝐴))) = (𝑁‘(1r‘𝑅))) |
26 | simp1 1132 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → 𝑅 ∈ NrmRing) | |
27 | 3, 7, 21 | nmmul 23275 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝐴 ∈ (Base‘𝑅) ∧ (𝐼‘𝐴) ∈ (Base‘𝑅)) → (𝑁‘(𝐴(.r‘𝑅)(𝐼‘𝐴))) = ((𝑁‘𝐴) · (𝑁‘(𝐼‘𝐴)))) |
28 | 26, 6, 16, 27 | syl3anc 1367 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘(𝐴(.r‘𝑅)(𝐼‘𝐴))) = ((𝑁‘𝐴) · (𝑁‘(𝐼‘𝐴)))) |
29 | 7, 22 | nm1 23278 | . . . 4 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) → (𝑁‘(1r‘𝑅)) = 1) |
30 | 29 | 3adant3 1128 | . . 3 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘(1r‘𝑅)) = 1) |
31 | 25, 28, 30 | 3eqtr3d 2866 | . 2 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → ((𝑁‘𝐴) · (𝑁‘(𝐼‘𝐴))) = 1) |
32 | 10, 19, 20, 31 | mvllmuld 11474 | 1 ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘(𝐼‘𝐴)) = (1 / (𝑁‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 1c1 10540 · cmul 10544 / cdiv 11299 Basecbs 16485 .rcmulr 16568 1rcur 19253 Ringcrg 19299 Unitcui 19391 invrcinvr 19423 NzRingcnzr 20032 normcnm 23188 NrmGrpcngp 23189 NrmRingcnrg 23191 |
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 |
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-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-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 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-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-0g 16717 df-topgen 16719 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-abv 19590 df-nzr 20033 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-xms 22932 df-ms 22933 df-nm 23194 df-ngp 23195 df-nrg 23197 |
This theorem is referenced by: nmdvr 23281 |
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