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Mirrors > Home > MPE Home > Th. List > tngnrg | Structured version Visualization version GIF version |
Description: Given any absolute value over a ring, augmenting the ring with the absolute value produces a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
tngnrg.t | ⊢ 𝑇 = (𝑅 toNrmGrp 𝐹) |
tngnrg.a | ⊢ 𝐴 = (AbsVal‘𝑅) |
Ref | Expression |
---|---|
tngnrg | ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngnrg.a | . . . . 5 ⊢ 𝐴 = (AbsVal‘𝑅) | |
2 | 1 | abvrcl 19594 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | ringgrp 19304 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Grp) |
5 | tngnrg.t | . . . . 5 ⊢ 𝑇 = (𝑅 toNrmGrp 𝐹) | |
6 | eqid 2823 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
7 | 5, 6 | tngds 23259 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ (-g‘𝑅)) = (dist‘𝑇)) |
8 | eqid 2823 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 1, 6 | abvmet 23187 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ (-g‘𝑅)) ∈ (Met‘(Base‘𝑅))) |
10 | 7, 9 | eqeltrrd 2916 | . . 3 ⊢ (𝐹 ∈ 𝐴 → (dist‘𝑇) ∈ (Met‘(Base‘𝑅))) |
11 | 1, 8 | abvf 19596 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹:(Base‘𝑅)⟶ℝ) |
12 | eqid 2823 | . . . . 5 ⊢ (dist‘𝑇) = (dist‘𝑇) | |
13 | 5, 8, 12 | tngngp2 23263 | . . . 4 ⊢ (𝐹:(Base‘𝑅)⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝑅 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘(Base‘𝑅))))) |
14 | 11, 13 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐴 → (𝑇 ∈ NrmGrp ↔ (𝑅 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘(Base‘𝑅))))) |
15 | 4, 10, 14 | mpbir2and 711 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmGrp) |
16 | reex 10630 | . . . . . 6 ⊢ ℝ ∈ V | |
17 | 5, 8, 16 | tngnm 23262 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝐹:(Base‘𝑅)⟶ℝ) → 𝐹 = (norm‘𝑇)) |
18 | 4, 11, 17 | syl2anc 586 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹 = (norm‘𝑇)) |
19 | eqidd 2824 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → (Base‘𝑅) = (Base‘𝑅)) | |
20 | 5, 8 | tngbas 23252 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → (Base‘𝑅) = (Base‘𝑇)) |
21 | eqid 2823 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
22 | 5, 21 | tngplusg 23253 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐴 → (+g‘𝑅) = (+g‘𝑇)) |
23 | 22 | oveqdr 7186 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑇)𝑦)) |
24 | eqid 2823 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
25 | 5, 24 | tngmulr 23255 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐴 → (.r‘𝑅) = (.r‘𝑇)) |
26 | 25 | oveqdr 7186 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑇)𝑦)) |
27 | 19, 20, 23, 26 | abvpropd 19615 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 → (AbsVal‘𝑅) = (AbsVal‘𝑇)) |
28 | 1, 27 | syl5eq 2870 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐴 = (AbsVal‘𝑇)) |
29 | 18, 28 | eleq12d 2909 | . . 3 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∈ 𝐴 ↔ (norm‘𝑇) ∈ (AbsVal‘𝑇))) |
30 | 29 | ibi 269 | . 2 ⊢ (𝐹 ∈ 𝐴 → (norm‘𝑇) ∈ (AbsVal‘𝑇)) |
31 | eqid 2823 | . . 3 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
32 | eqid 2823 | . . 3 ⊢ (AbsVal‘𝑇) = (AbsVal‘𝑇) | |
33 | 31, 32 | isnrg 23271 | . 2 ⊢ (𝑇 ∈ NrmRing ↔ (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) ∈ (AbsVal‘𝑇))) |
34 | 15, 30, 33 | sylanbrc 585 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 distcds 16576 Grpcgrp 18105 -gcsg 18107 Ringcrg 19299 AbsValcabv 19589 Metcmet 20533 normcnm 23188 NrmGrpcngp 23189 toNrmGrp ctng 23190 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-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-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-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-seq 13373 df-exp 13433 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-mulr 16581 df-tset 16586 df-ds 16589 df-rest 16698 df-topn 16699 df-0g 16717 df-topgen 16719 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mgp 19242 df-ur 19254 df-ring 19301 df-abv 19590 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-tng 23196 df-nrg 23197 |
This theorem is referenced by: (None) |
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