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Mirrors > Home > MPE Home > Th. List > subrg0 | Structured version Visualization version GIF version |
Description: A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
subrg0.1 | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
subrg0.2 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
subrg0 | ⊢ (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgsubg 20157 | . 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅)) | |
2 | subrg0.1 | . . 3 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
3 | subrg0.2 | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | 2, 3 | subg0 18869 | . 2 ⊢ (𝐴 ∈ (SubGrp‘𝑅) → 0 = (0g‘𝑆)) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6492 (class class class)co 7350 ↾s cress 17048 0gc0g 17257 SubGrpcsubg 18857 SubRingcsubrg 20147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-ress 17049 df-plusg 17082 df-0g 17259 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-grp 18687 df-subg 18860 df-ring 19896 df-subrg 20149 |
This theorem is referenced by: issubdrg 20176 subdrgint 20199 subrgnzr 20667 qsssubdrg 20785 ressmplbas2 21356 subrgmvr 21362 subrgascl 21402 clm0 24363 cphreccllem 24470 sdrgdvcl 31876 sdrginvcl 31877 evls1fpws 32065 ressdeg1 32068 ressply10g 32070 fedgmullem1 32114 fedgmullem2 32115 evlsbagval 40663 |
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