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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 2824 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
4 | eqid 2824 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | 1, 2, 3, 4 | rrgval 20063 | . 2 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → 𝑦 = (0g‘𝑅))} |
6 | 5 | ssrab3 4060 | 1 ⊢ 𝐸 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∀wral 3141 ⊆ wss 3939 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 .rcmulr 16569 0gc0g 16716 RLRegcrlreg 20055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-rlreg 20059 |
This theorem is referenced by: znrrg 20715 mdegvsca 24673 deg1mul3 24712 |
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