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Mirrors > Home > MPE Home > Th. List > mplval2 | Structured version Visualization version GIF version |
Description: Self-referential expression for the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
mplval2.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mplval2.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
mplval2.u | ⊢ 𝑈 = (Base‘𝑃) |
Ref | Expression |
---|---|
mplval2 | ⊢ 𝑃 = (𝑆 ↾s 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mplval2.p | . 2 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
2 | mplval2.s | . 2 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
3 | eqid 2821 | . 2 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
4 | eqid 2821 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | mplval2.u | . . 3 ⊢ 𝑈 = (Base‘𝑃) | |
6 | 1, 2, 3, 4, 5 | mplbas 20203 | . 2 ⊢ 𝑈 = {𝑓 ∈ (Base‘𝑆) ∣ 𝑓 finSupp (0g‘𝑅)} |
7 | 1, 2, 3, 4, 6 | mplval 20202 | 1 ⊢ 𝑃 = (𝑆 ↾s 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 ↾s cress 16478 0gc0g 16707 mPwSer cmps 20125 mPoly cmpl 20127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-1cn 10589 ax-addcl 10591 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-nn 11633 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-psr 20130 df-mpl 20132 |
This theorem is referenced by: mpl0 20215 mpladd 20216 mplmul 20217 mpl1 20218 mplsca 20219 mplvsca2 20220 mplgrp 20224 mpllmod 20225 mplring 20226 mplcrng 20228 mplassa 20229 ressmpladd 20232 ressmplmul 20233 ressmplvsca 20234 subrgmpl 20235 mplbas2 20245 mplind 20276 evlseu 20290 mplplusg 20382 mplmulr 20383 |
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