Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sraring | Structured version Visualization version GIF version |
Description: Condition for a subring algebra to be a ring. (Contributed by Thierry Arnoux, 24-Jul-2023.) |
Ref | Expression |
---|---|
sraring.1 | ⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉) |
sraring.2 | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
sraring | ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sraring.2 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑉 ⊆ 𝐵 → 𝐵 = (Base‘𝑅)) |
3 | sraring.1 | . . . . . 6 ⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉) | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝑉 ⊆ 𝐵 → 𝐴 = ((subringAlg ‘𝑅)‘𝑉)) |
5 | id 22 | . . . . . 6 ⊢ (𝑉 ⊆ 𝐵 → 𝑉 ⊆ 𝐵) | |
6 | 5, 1 | sseqtrdi 3976 | . . . . 5 ⊢ (𝑉 ⊆ 𝐵 → 𝑉 ⊆ (Base‘𝑅)) |
7 | 4, 6 | srabase 20452 | . . . 4 ⊢ (𝑉 ⊆ 𝐵 → (Base‘𝑅) = (Base‘𝐴)) |
8 | 1, 7 | eqtrid 2792 | . . 3 ⊢ (𝑉 ⊆ 𝐵 → 𝐵 = (Base‘𝐴)) |
9 | 4, 6 | sraaddg 20454 | . . . 4 ⊢ (𝑉 ⊆ 𝐵 → (+g‘𝑅) = (+g‘𝐴)) |
10 | 9 | oveqdr 7300 | . . 3 ⊢ ((𝑉 ⊆ 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝐴)𝑦)) |
11 | 4, 6 | sramulr 20456 | . . . 4 ⊢ (𝑉 ⊆ 𝐵 → (.r‘𝑅) = (.r‘𝐴)) |
12 | 11 | oveqdr 7300 | . . 3 ⊢ ((𝑉 ⊆ 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝐴)𝑦)) |
13 | 2, 8, 10, 12 | ringpropd 19832 | . 2 ⊢ (𝑉 ⊆ 𝐵 → (𝑅 ∈ Ring ↔ 𝐴 ∈ Ring)) |
14 | 13 | biimpac 479 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 ‘cfv 6432 Basecbs 16923 +gcplusg 16973 .rcmulr 16974 Ringcrg 19794 subringAlg csra 20441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-plusg 16986 df-mulr 16987 df-sca 16989 df-vsca 16990 df-ip 16991 df-0g 17163 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-grp 18591 df-mgp 19732 df-ring 19796 df-sra 20445 |
This theorem is referenced by: sradrng 31682 rgmoddim 31702 fedgmullem2 31720 ccfldsrarelvec 31750 |
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