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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 2734 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | eqid 2734 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | 1, 2, 3, 4 | rrgval 20714 | . 2 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → 𝑦 = (0g‘𝑅))} |
6 | 5 | ssrab3 4099 | 1 ⊢ 𝐸 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∀wral 3063 ⊆ wss 3970 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 .rcmulr 17307 0gc0g 17494 RLRegcrlreg 20708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-iota 6524 df-fun 6574 df-fv 6580 df-ov 7448 df-rlreg 20711 |
This theorem is referenced by: isdomn6 20731 znrrg 21602 mdegvsca 26127 deg1mul3 26167 rrgsubm 33245 fracbas 33264 fracerl 33265 fracfld 33267 assalactf1o 33640 assarrginv 33641 |
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