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Mirrors > Home > MPE Home > Th. List > 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 4046 | . . . . 5 ⊢ (𝑉 ⊆ 𝐵 → 𝑉 ⊆ (Base‘𝑅)) |
7 | 4, 6 | srabase 21195 | . . . 4 ⊢ (𝑉 ⊆ 𝐵 → (Base‘𝑅) = (Base‘𝐴)) |
8 | 1, 7 | eqtrid 2787 | . . 3 ⊢ (𝑉 ⊆ 𝐵 → 𝐵 = (Base‘𝐴)) |
9 | 4, 6 | sraaddg 21197 | . . . 4 ⊢ (𝑉 ⊆ 𝐵 → (+g‘𝑅) = (+g‘𝐴)) |
10 | 9 | oveqdr 7459 | . . 3 ⊢ ((𝑉 ⊆ 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝐴)𝑦)) |
11 | 4, 6 | sramulr 21199 | . . . 4 ⊢ (𝑉 ⊆ 𝐵 → (.r‘𝑅) = (.r‘𝐴)) |
12 | 11 | oveqdr 7459 | . . 3 ⊢ ((𝑉 ⊆ 𝐵 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝐴)𝑦)) |
13 | 2, 8, 10, 12 | ringpropd 20302 | . 2 ⊢ (𝑉 ⊆ 𝐵 → (𝑅 ∈ Ring ↔ 𝐴 ∈ Ring)) |
14 | 13 | biimpac 478 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ‘cfv 6563 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Ringcrg 20251 subringAlg csra 21188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-mgp 20153 df-ring 20253 df-sra 21190 |
This theorem is referenced by: sraassab 21906 sradrng 33613 rgmoddimOLD 33638 fedgmullem2 33658 ccfldsrarelvec 33696 |
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