| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cnzh | Structured version Visualization version GIF version | ||
| Description: The ℤ-module of ℂ is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.) |
| Ref | Expression |
|---|---|
| cnzh | ⊢ (ℤMod‘ℂfld) ∈ NrmMod |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnnrg 24801 | . . . 4 ⊢ ℂfld ∈ NrmRing | |
| 2 | eqid 2737 | . . . . 5 ⊢ (ℤMod‘ℂfld) = (ℤMod‘ℂfld) | |
| 3 | 2 | zhmnrg 33966 | . . . 4 ⊢ (ℂfld ∈ NrmRing → (ℤMod‘ℂfld) ∈ NrmRing) |
| 4 | nrgngp 24683 | . . . 4 ⊢ ((ℤMod‘ℂfld) ∈ NrmRing → (ℤMod‘ℂfld) ∈ NrmGrp) | |
| 5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ NrmGrp |
| 6 | nrgring 24684 | . . . . 5 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ Ring) | |
| 7 | ringabl 20278 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Abel) | |
| 8 | 1, 6, 7 | mp2b 10 | . . . 4 ⊢ ℂfld ∈ Abel |
| 9 | 2 | zlmlmod 21537 | . . . 4 ⊢ (ℂfld ∈ Abel ↔ (ℤMod‘ℂfld) ∈ LMod) |
| 10 | 8, 9 | mpbi 230 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ LMod |
| 11 | zringnrg 24809 | . . 3 ⊢ ℤring ∈ NrmRing | |
| 12 | 5, 10, 11 | 3pm3.2i 1340 | . 2 ⊢ ((ℤMod‘ℂfld) ∈ NrmGrp ∧ (ℤMod‘ℂfld) ∈ LMod ∧ ℤring ∈ NrmRing) |
| 13 | simpl 482 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑧 ∈ ℤ) | |
| 14 | 13 | zcnd 12723 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑧 ∈ ℂ) |
| 15 | simpr 484 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
| 16 | 14, 15 | absmuld 15493 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧 · 𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
| 17 | cnfldmulg 21416 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧 · 𝑥)) | |
| 18 | 17 | fveq2d 6910 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧(.g‘ℂfld)𝑥)) = (abs‘(𝑧 · 𝑥))) |
| 19 | fvres 6925 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) | |
| 20 | 19 | adantr 480 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) |
| 21 | 20 | oveq1d 7446 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
| 22 | 16, 18, 21 | 3eqtr4d 2787 | . . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥))) |
| 23 | 22 | rgen2 3199 | . 2 ⊢ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℂ (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥)) |
| 24 | cnfldbas 21368 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
| 25 | 2, 24 | zlmbas 21529 | . . 3 ⊢ ℂ = (Base‘(ℤMod‘ℂfld)) |
| 26 | cnfldex 21367 | . . . 4 ⊢ ℂfld ∈ V | |
| 27 | cnfldnm 24799 | . . . . 5 ⊢ abs = (norm‘ℂfld) | |
| 28 | 2, 27 | zlmnm 33965 | . . . 4 ⊢ (ℂfld ∈ V → abs = (norm‘(ℤMod‘ℂfld))) |
| 29 | 26, 28 | ax-mp 5 | . . 3 ⊢ abs = (norm‘(ℤMod‘ℂfld)) |
| 30 | eqid 2737 | . . . 4 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
| 31 | 2, 30 | zlmvsca 21536 | . . 3 ⊢ (.g‘ℂfld) = ( ·𝑠 ‘(ℤMod‘ℂfld)) |
| 32 | 2 | zlmsca 21535 | . . . 4 ⊢ (ℂfld ∈ V → ℤring = (Scalar‘(ℤMod‘ℂfld))) |
| 33 | 26, 32 | ax-mp 5 | . . 3 ⊢ ℤring = (Scalar‘(ℤMod‘ℂfld)) |
| 34 | zringbas 21464 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
| 35 | zringnm 33957 | . . . 4 ⊢ (norm‘ℤring) = (abs ↾ ℤ) | |
| 36 | 35 | eqcomi 2746 | . . 3 ⊢ (abs ↾ ℤ) = (norm‘ℤring) |
| 37 | 25, 29, 31, 33, 34, 36 | isnlm 24696 | . 2 ⊢ ((ℤMod‘ℂfld) ∈ NrmMod ↔ (((ℤMod‘ℂfld) ∈ NrmGrp ∧ (ℤMod‘ℂfld) ∈ LMod ∧ ℤring ∈ NrmRing) ∧ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℂ (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥)))) |
| 38 | 12, 23, 37 | mpbir2an 711 | 1 ⊢ (ℤMod‘ℂfld) ∈ NrmMod |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 · cmul 11160 ℤcz 12613 abscabs 15273 Scalarcsca 17300 .gcmg 19085 Abelcabl 19799 Ringcrg 20230 LModclmod 20858 ℂfldccnfld 21364 ℤringczring 21457 ℤModczlm 21511 normcnm 24589 NrmGrpcngp 24590 NrmRingcnrg 24592 NrmModcnlm 24593 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-0g 17486 df-topgen 17488 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-abv 20810 df-lmod 20860 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-zring 21458 df-zlm 21515 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-xms 24330 df-ms 24331 df-nm 24595 df-ngp 24596 df-nrg 24598 df-nlm 24599 |
| This theorem is referenced by: cnrrext 34011 |
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