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Mirrors > Home > MPE Home > Th. List > nmgt0 | Structured version Visualization version GIF version |
Description: The norm of a nonzero element is a positive real. (Contributed by NM, 20-Nov-2007.) (Revised by AV, 8-Oct-2021.) |
Ref | Expression |
---|---|
nmgt0.x | ⊢ 𝑋 = (Base‘𝐺) |
nmgt0.n | ⊢ 𝑁 = (norm‘𝐺) |
nmgt0.z | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
nmgt0 | ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝐴 ≠ 0 ↔ 0 < (𝑁‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmgt0.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
2 | nmgt0.n | . . . 4 ⊢ 𝑁 = (norm‘𝐺) | |
3 | nmgt0.z | . . . 4 ⊢ 0 = (0g‘𝐺) | |
4 | 1, 2, 3 | nmeq0 22512 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = 0 ↔ 𝐴 = 0 )) |
5 | 4 | necon3bid 2908 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0 )) |
6 | 1, 2 | nmcl 22510 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
7 | 1, 2 | nmge0 22511 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝑁‘𝐴)) |
8 | ne0gt0 10223 | . . 3 ⊢ (((𝑁‘𝐴) ∈ ℝ ∧ 0 ≤ (𝑁‘𝐴)) → ((𝑁‘𝐴) ≠ 0 ↔ 0 < (𝑁‘𝐴))) | |
9 | 6, 7, 8 | syl2anc 696 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) ≠ 0 ↔ 0 < (𝑁‘𝐴))) |
10 | 5, 9 | bitr3d 270 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝐴 ≠ 0 ↔ 0 < (𝑁‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1564 ∈ wcel 2071 ≠ wne 2864 class class class wbr 4728 ‘cfv 5969 ℝcr 10016 0cc0 10017 < clt 10155 ≤ cle 10156 Basecbs 15948 0gc0g 16191 normcnm 22471 NrmGrpcngp 22472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 ax-cnex 10073 ax-resscn 10074 ax-1cn 10075 ax-icn 10076 ax-addcl 10077 ax-addrcl 10078 ax-mulcl 10079 ax-mulrcl 10080 ax-mulcom 10081 ax-addass 10082 ax-mulass 10083 ax-distr 10084 ax-i2m1 10085 ax-1ne0 10086 ax-1rid 10087 ax-rnegex 10088 ax-rrecex 10089 ax-cnre 10090 ax-pre-lttri 10091 ax-pre-lttrn 10092 ax-pre-ltadd 10093 ax-pre-mulgt0 10094 ax-pre-sup 10095 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-nel 2968 df-ral 2987 df-rex 2988 df-reu 2989 df-rmo 2990 df-rab 2991 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-pss 3664 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-tp 4258 df-op 4260 df-uni 4513 df-iun 4598 df-br 4729 df-opab 4789 df-mpt 4806 df-tr 4829 df-id 5096 df-eprel 5101 df-po 5107 df-so 5108 df-fr 5145 df-we 5147 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-pred 5761 df-ord 5807 df-on 5808 df-lim 5809 df-suc 5810 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-f1 5974 df-fo 5975 df-f1o 5976 df-fv 5977 df-riota 6694 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-om 7151 df-1st 7253 df-2nd 7254 df-wrecs 7495 df-recs 7556 df-rdg 7594 df-er 7830 df-map 7944 df-en 8041 df-dom 8042 df-sdom 8043 df-sup 8432 df-inf 8433 df-pnf 10157 df-mnf 10158 df-xr 10159 df-ltxr 10160 df-le 10161 df-sub 10349 df-neg 10350 df-div 10766 df-nn 11102 df-2 11160 df-n0 11374 df-z 11459 df-uz 11769 df-q 11871 df-rp 11915 df-xneg 12028 df-xadd 12029 df-xmul 12030 df-0g 16193 df-topgen 16195 df-mgm 17332 df-sgrp 17374 df-mnd 17385 df-grp 17515 df-psmet 19829 df-xmet 19830 df-met 19831 df-bl 19832 df-mopn 19833 df-top 20790 df-topon 20807 df-topsp 20828 df-bases 20841 df-xms 22215 df-ms 22216 df-nm 22477 df-ngp 22478 |
This theorem is referenced by: ncvs1 23046 |
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