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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 21528 | . 2 ⊢ Rel dom mPoly | |
4 | 1, 2, 3 | strov2rcl 17147 | 1 ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ‘cfv 6539 (class class class)co 7403 Basecbs 17139 mPoly cmpl 21440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-1cn 11163 ax-addcl 11165 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-nn 12208 df-slot 17110 df-ndx 17122 df-base 17140 df-mpl 21445 |
This theorem is referenced by: mdegldg 25565 |
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