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Mirrors > Home > MPE Home > Th. List > zsssubrg | Structured version Visualization version GIF version |
Description: The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
zsssubrg | ⊢ (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 478 | . . . . . 6 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ) | |
2 | ax-1cn 10283 | . . . . . 6 ⊢ 1 ∈ ℂ | |
3 | cnfldmulg 20099 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 1 ∈ ℂ) → (𝑥(.g‘ℂfld)1) = (𝑥 · 1)) | |
4 | 1, 2, 3 | sylancl 581 | . . . . 5 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥 ∈ ℤ) → (𝑥(.g‘ℂfld)1) = (𝑥 · 1)) |
5 | zcn 11670 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
6 | 5 | adantl 474 | . . . . . 6 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℂ) |
7 | 6 | mulid1d 10347 | . . . . 5 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥 ∈ ℤ) → (𝑥 · 1) = 𝑥) |
8 | 4, 7 | eqtrd 2834 | . . . 4 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥 ∈ ℤ) → (𝑥(.g‘ℂfld)1) = 𝑥) |
9 | subrgsubg 19103 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → 𝑅 ∈ (SubGrp‘ℂfld)) | |
10 | 9 | adantr 473 | . . . . 5 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥 ∈ ℤ) → 𝑅 ∈ (SubGrp‘ℂfld)) |
11 | cnfld1 20092 | . . . . . . 7 ⊢ 1 = (1r‘ℂfld) | |
12 | 11 | subrg1cl 19105 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → 1 ∈ 𝑅) |
13 | 12 | adantr 473 | . . . . 5 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥 ∈ ℤ) → 1 ∈ 𝑅) |
14 | eqid 2800 | . . . . . 6 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
15 | 14 | subgmulgcl 17919 | . . . . 5 ⊢ ((𝑅 ∈ (SubGrp‘ℂfld) ∧ 𝑥 ∈ ℤ ∧ 1 ∈ 𝑅) → (𝑥(.g‘ℂfld)1) ∈ 𝑅) |
16 | 10, 1, 13, 15 | syl3anc 1491 | . . . 4 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥 ∈ ℤ) → (𝑥(.g‘ℂfld)1) ∈ 𝑅) |
17 | 8, 16 | eqeltrrd 2880 | . . 3 ⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ 𝑅) |
18 | 17 | ex 402 | . 2 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → (𝑥 ∈ ℤ → 𝑥 ∈ 𝑅)) |
19 | 18 | ssrdv 3805 | 1 ⊢ (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ⊆ wss 3770 ‘cfv 6102 (class class class)co 6879 ℂcc 10223 1c1 10226 · cmul 10230 ℤcz 11665 .gcmg 17855 SubGrpcsubg 17900 SubRingcsubrg 19093 ℂfldccnfld 20067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-inf2 8789 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 ax-addf 10304 ax-mulf 10305 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-oadd 7804 df-er 7983 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-7 11380 df-8 11381 df-9 11382 df-n0 11580 df-z 11666 df-dec 11783 df-uz 11930 df-fz 12580 df-seq 13055 df-struct 16185 df-ndx 16186 df-slot 16187 df-base 16189 df-sets 16190 df-ress 16191 df-plusg 16279 df-mulr 16280 df-starv 16281 df-tset 16285 df-ple 16286 df-ds 16288 df-unif 16289 df-0g 16416 df-mgm 17556 df-sgrp 17598 df-mnd 17609 df-grp 17740 df-minusg 17741 df-mulg 17856 df-subg 17903 df-cmn 18509 df-mgp 18805 df-ur 18817 df-ring 18864 df-cring 18865 df-subrg 19095 df-cnfld 20068 |
This theorem is referenced by: qsssubdrg 20126 clmzss 23204 dvply2g 24380 |
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