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Mirrors > Home > MPE Home > Th. List > mplrcl | Structured version Visualization version GIF version |
Description: Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
mplrcl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplrcl.b | ⊢ 𝐵 = (Base‘𝑃) |
Ref | Expression |
---|---|
mplrcl | ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplrcl.p | . 2 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
2 | mplrcl.b | . 2 ⊢ 𝐵 = (Base‘𝑃) | |
3 | reldmmpl 19248 | . 2 ⊢ Rel dom mPoly | |
4 | 1, 2, 3 | strov2rcl 15750 | 1 ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 mPoly cmpl 19174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-slot 15699 df-base 15700 df-mpl 19179 |
This theorem is referenced by: mdegleb 23628 mdeglt 23629 mdegldg 23630 mdegxrcl 23631 mdegcl 23633 mdegnn0cl 23635 |
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