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Mirrors > Home > MPE Home > Th. List > subrngmcl | Structured version Visualization version GIF version |
Description: A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 20517. (Revised by AV, 14-Feb-2025.) |
Ref | Expression |
---|---|
subrngmcl.p | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
subrngmcl | ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . . 5 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
2 | 1 | subrngrng 20481 | . . . 4 ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝑅 ↾s 𝐴) ∈ Rng) |
3 | 2 | 3ad2ant1 1131 | . . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑅 ↾s 𝐴) ∈ Rng) |
4 | simp2 1135 | . . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
5 | 1 | subrngbas 20485 | . . . . 5 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘(𝑅 ↾s 𝐴))) |
6 | 5 | 3ad2ant1 1131 | . . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝐴 = (Base‘(𝑅 ↾s 𝐴))) |
7 | 4, 6 | eleqtrd 2831 | . . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ (Base‘(𝑅 ↾s 𝐴))) |
8 | simp3 1136 | . . . 4 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ 𝐴) | |
9 | 8, 6 | eleqtrd 2831 | . . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ (Base‘(𝑅 ↾s 𝐴))) |
10 | eqid 2728 | . . . 4 ⊢ (Base‘(𝑅 ↾s 𝐴)) = (Base‘(𝑅 ↾s 𝐴)) | |
11 | eqid 2728 | . . . 4 ⊢ (.r‘(𝑅 ↾s 𝐴)) = (.r‘(𝑅 ↾s 𝐴)) | |
12 | 10, 11 | rngcl 20098 | . . 3 ⊢ (((𝑅 ↾s 𝐴) ∈ Rng ∧ 𝑋 ∈ (Base‘(𝑅 ↾s 𝐴)) ∧ 𝑌 ∈ (Base‘(𝑅 ↾s 𝐴))) → (𝑋(.r‘(𝑅 ↾s 𝐴))𝑌) ∈ (Base‘(𝑅 ↾s 𝐴))) |
13 | 3, 7, 9, 12 | syl3anc 1369 | . 2 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋(.r‘(𝑅 ↾s 𝐴))𝑌) ∈ (Base‘(𝑅 ↾s 𝐴))) |
14 | subrngmcl.p | . . . . 5 ⊢ · = (.r‘𝑅) | |
15 | 1, 14 | ressmulr 17282 | . . . 4 ⊢ (𝐴 ∈ (SubRng‘𝑅) → · = (.r‘(𝑅 ↾s 𝐴))) |
16 | 15 | 3ad2ant1 1131 | . . 3 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → · = (.r‘(𝑅 ↾s 𝐴))) |
17 | 16 | oveqd 7432 | . 2 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) = (𝑋(.r‘(𝑅 ↾s 𝐴))𝑌)) |
18 | 13, 17, 6 | 3eltr4d 2844 | 1 ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 ↾s cress 17203 .rcmulr 17228 Rngcrng 20086 SubRngcsubrng 20476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-mgm 18594 df-sgrp 18673 df-subg 19072 df-abl 19732 df-mgp 20069 df-rng 20087 df-subrng 20477 |
This theorem is referenced by: issubrng2 20489 subrngint 20491 rhmimasubrnglem 20496 rhmimasubrng 20497 subrgmcl 20517 |
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