Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 20135 | . 2 ⊢ Rel dom mPoly | |
4 | 1, 2, 3 | strov2rcl 16534 | 1 ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 mPoly cmpl 20061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-slot 16475 df-base 16477 df-mpl 20066 |
This theorem is referenced by: mdegleb 24585 mdeglt 24586 mdegldg 24587 mdegxrcl 24588 mdegcl 24590 mdegnn0cl 24592 |
Copyright terms: Public domain | W3C validator |