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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 24817 | . . . 4 ⊢ ℂfld ∈ NrmRing | |
2 | eqid 2735 | . . . . 5 ⊢ (ℤMod‘ℂfld) = (ℤMod‘ℂfld) | |
3 | 2 | zhmnrg 33928 | . . . 4 ⊢ (ℂfld ∈ NrmRing → (ℤMod‘ℂfld) ∈ NrmRing) |
4 | nrgngp 24699 | . . . 4 ⊢ ((ℤMod‘ℂfld) ∈ NrmRing → (ℤMod‘ℂfld) ∈ NrmGrp) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ NrmGrp |
6 | nrgring 24700 | . . . . 5 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ Ring) | |
7 | ringabl 20295 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Abel) | |
8 | 1, 6, 7 | mp2b 10 | . . . 4 ⊢ ℂfld ∈ Abel |
9 | 2 | zlmlmod 21555 | . . . 4 ⊢ (ℂfld ∈ Abel ↔ (ℤMod‘ℂfld) ∈ LMod) |
10 | 8, 9 | mpbi 230 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ LMod |
11 | zringnrg 24824 | . . 3 ⊢ ℤring ∈ NrmRing | |
12 | 5, 10, 11 | 3pm3.2i 1338 | . 2 ⊢ ((ℤMod‘ℂfld) ∈ NrmGrp ∧ (ℤMod‘ℂfld) ∈ LMod ∧ ℤring ∈ NrmRing) |
13 | simpl 482 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑧 ∈ ℤ) | |
14 | 13 | zcnd 12721 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑧 ∈ ℂ) |
15 | simpr 484 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
16 | 14, 15 | absmuld 15490 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧 · 𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
17 | cnfldmulg 21434 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧 · 𝑥)) | |
18 | 17 | fveq2d 6911 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧(.g‘ℂfld)𝑥)) = (abs‘(𝑧 · 𝑥))) |
19 | fvres 6926 | . . . . . 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 2785 | . . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥))) |
23 | 22 | rgen2 3197 | . 2 ⊢ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℂ (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥)) |
24 | cnfldbas 21386 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
25 | 2, 24 | zlmbas 21547 | . . 3 ⊢ ℂ = (Base‘(ℤMod‘ℂfld)) |
26 | cnfldex 21385 | . . . 4 ⊢ ℂfld ∈ V | |
27 | cnfldnm 24815 | . . . . 5 ⊢ abs = (norm‘ℂfld) | |
28 | 2, 27 | zlmnm 33927 | . . . 4 ⊢ (ℂfld ∈ V → abs = (norm‘(ℤMod‘ℂfld))) |
29 | 26, 28 | ax-mp 5 | . . 3 ⊢ abs = (norm‘(ℤMod‘ℂfld)) |
30 | eqid 2735 | . . . 4 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
31 | 2, 30 | zlmvsca 21554 | . . 3 ⊢ (.g‘ℂfld) = ( ·𝑠 ‘(ℤMod‘ℂfld)) |
32 | 2 | zlmsca 21553 | . . . 4 ⊢ (ℂfld ∈ V → ℤring = (Scalar‘(ℤMod‘ℂfld))) |
33 | 26, 32 | ax-mp 5 | . . 3 ⊢ ℤring = (Scalar‘(ℤMod‘ℂfld)) |
34 | zringbas 21482 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
35 | zringnm 33919 | . . . 4 ⊢ (norm‘ℤring) = (abs ↾ ℤ) | |
36 | 35 | eqcomi 2744 | . . 3 ⊢ (abs ↾ ℤ) = (norm‘ℤring) |
37 | 25, 29, 31, 33, 34, 36 | isnlm 24712 | . 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 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 · cmul 11158 ℤcz 12611 abscabs 15270 Scalarcsca 17301 .gcmg 19098 Abelcabl 19814 Ringcrg 20251 LModclmod 20875 ℂfldccnfld 21382 ℤringczring 21475 ℤModczlm 21529 normcnm 24605 NrmGrpcngp 24606 NrmRingcnrg 24608 NrmModcnlm 24609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ico 13390 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17469 df-topn 17470 df-0g 17488 df-topgen 17490 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-abv 20827 df-lmod 20877 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-zring 21476 df-zlm 21533 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-xms 24346 df-ms 24347 df-nm 24611 df-ngp 24612 df-nrg 24614 df-nlm 24615 |
This theorem is referenced by: cnrrext 33973 |
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