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Mirrors > Home > MPE Home > Th. List > zring0 | Structured version Visualization version GIF version |
Description: The zero element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.) |
Ref | Expression |
---|---|
zring0 | ⊢ 0 = (0g‘ℤring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncrng 21419 | . . 3 ⊢ ℂfld ∈ CRing | |
2 | crngring 20267 | . . 3 ⊢ (ℂfld ∈ CRing → ℂfld ∈ Ring) | |
3 | ringmnd 20265 | . . 3 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ ℂfld ∈ Mnd |
5 | 0z 12646 | . 2 ⊢ 0 ∈ ℤ | |
6 | zsscn 12643 | . 2 ⊢ ℤ ⊆ ℂ | |
7 | df-zring 21476 | . . 3 ⊢ ℤring = (ℂfld ↾s ℤ) | |
8 | cnfldbas 21386 | . . 3 ⊢ ℂ = (Base‘ℂfld) | |
9 | cnfld0 21423 | . . 3 ⊢ 0 = (0g‘ℂfld) | |
10 | 7, 8, 9 | ress0g 18795 | . 2 ⊢ ((ℂfld ∈ Mnd ∧ 0 ∈ ℤ ∧ ℤ ⊆ ℂ) → 0 = (0g‘ℤring)) |
11 | 4, 5, 6, 10 | mp3an 1461 | 1 ⊢ 0 = (0g‘ℤring) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2103 ⊆ wss 3970 ‘cfv 6572 ℂcc 11178 0cc0 11180 ℤcz 12635 0gc0g 17494 Mndcmnd 18767 Ringcrg 20255 CRingccrg 20256 ℂfldccnfld 21382 ℤringczring 21475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-addf 11259 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-fz 13564 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-0g 17496 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-grp 18971 df-cmn 19819 df-mgp 20157 df-ring 20257 df-cring 20258 df-cnfld 21383 df-zring 21476 |
This theorem is referenced by: zringnzr 21489 zringlpirlem1 21491 zringinvg 21494 zringlpir 21496 zringndrg 21497 prmirred 21503 zrh0 21542 zndvds0 21587 lgseisenlem4 27431 zringidom 33536 zringfrac 33539 zrhf1ker 33913 zlmodzxz0 47999 zlmodzxzldep 48151 |
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