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Mirrors > Home > MPE Home > Th. List > rrgss | Structured version Visualization version GIF version |
Description: Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
Ref | Expression |
---|---|
rrgss.e | ⊢ 𝐸 = (RLReg‘𝑅) |
rrgss.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
rrgss | ⊢ 𝐸 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrgss.e | . . 3 ⊢ 𝐸 = (RLReg‘𝑅) | |
2 | rrgss.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | eqid 2740 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | eqid 2740 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | 1, 2, 3, 4 | rrgval 20548 | . 2 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → 𝑦 = (0g‘𝑅))} |
6 | 5 | ssrab3 4020 | 1 ⊢ 𝐸 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∀wral 3066 ⊆ wss 3892 ‘cfv 6431 (class class class)co 7269 Basecbs 16902 .rcmulr 16953 0gc0g 17140 RLRegcrlreg 20540 |
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-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 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-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6389 df-fun 6433 df-fv 6439 df-ov 7272 df-rlreg 20544 |
This theorem is referenced by: znrrg 20763 mdegvsca 25231 deg1mul3 25270 |
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