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Mirrors > Home > MPE Home > Th. List > evlvar | Structured version Visualization version GIF version |
Description: Simple polynomial evaluation maps variables to projections. (Contributed by AV, 12-Sep-2019.) |
Ref | Expression |
---|---|
evlvar.q | ⊢ 𝑄 = (𝐼 eval 𝑆) |
evlvar.v | ⊢ 𝑉 = (𝐼 mVar 𝑆) |
evlvar.b | ⊢ 𝐵 = (Base‘𝑆) |
evlvar.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
evlvar.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlvar.x | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
Ref | Expression |
---|---|
evlvar | ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ ((𝐼 evalSub 𝑆)‘𝐵) = ((𝐼 evalSub 𝑆)‘𝐵) | |
2 | evlvar.q | . . 3 ⊢ 𝑄 = (𝐼 eval 𝑆) | |
3 | eqid 2738 | . . 3 ⊢ (𝐼 mVar (𝑆 ↾s 𝐵)) = (𝐼 mVar (𝑆 ↾s 𝐵)) | |
4 | eqid 2738 | . . 3 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
5 | evlvar.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
6 | evlvar.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
7 | evlvar.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
8 | crngring 19783 | . . . 4 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
9 | 5 | subrgid 20014 | . . . 4 ⊢ (𝑆 ∈ Ring → 𝐵 ∈ (SubRing‘𝑆)) |
10 | 7, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑆)) |
11 | evlvar.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
12 | 1, 2, 3, 4, 5, 6, 7, 10, 11 | evlsvarsrng 21297 | . 2 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘((𝐼 mVar (𝑆 ↾s 𝐵))‘𝑋)) = (𝑄‘((𝐼 mVar (𝑆 ↾s 𝐵))‘𝑋))) |
13 | 1, 3, 4, 5, 6, 7, 10, 11 | evlsvar 21288 | . 2 ⊢ (𝜑 → (((𝐼 evalSub 𝑆)‘𝐵)‘((𝐼 mVar (𝑆 ↾s 𝐵))‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) |
14 | evlvar.v | . . . . . 6 ⊢ 𝑉 = (𝐼 mVar 𝑆) | |
15 | 14, 6, 10, 4 | subrgmvr 21222 | . . . . 5 ⊢ (𝜑 → 𝑉 = (𝐼 mVar (𝑆 ↾s 𝐵))) |
16 | 15 | fveq1d 6769 | . . . 4 ⊢ (𝜑 → (𝑉‘𝑋) = ((𝐼 mVar (𝑆 ↾s 𝐵))‘𝑋)) |
17 | 16 | eqcomd 2744 | . . 3 ⊢ (𝜑 → ((𝐼 mVar (𝑆 ↾s 𝐵))‘𝑋) = (𝑉‘𝑋)) |
18 | 17 | fveq2d 6771 | . 2 ⊢ (𝜑 → (𝑄‘((𝐼 mVar (𝑆 ↾s 𝐵))‘𝑋)) = (𝑄‘(𝑉‘𝑋))) |
19 | 12, 13, 18 | 3eqtr3rd 2787 | 1 ⊢ (𝜑 → (𝑄‘(𝑉‘𝑋)) = (𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑔‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ↦ cmpt 5157 ‘cfv 6427 (class class class)co 7268 ↑m cmap 8603 Basecbs 16900 ↾s cress 16929 Ringcrg 19771 CRingccrg 19772 SubRingcsubrg 20008 mVar cmvr 21096 evalSub ces 21268 eval cevl 21269 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-isom 6436 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-ofr 7525 df-om 7704 df-1st 7821 df-2nd 7822 df-supp 7966 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-map 8605 df-pm 8606 df-ixp 8674 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-fsupp 9117 df-sup 9189 df-oi 9257 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-2 12024 df-3 12025 df-4 12026 df-5 12027 df-6 12028 df-7 12029 df-8 12030 df-9 12031 df-n0 12222 df-z 12308 df-dec 12426 df-uz 12571 df-fz 13228 df-fzo 13371 df-seq 13710 df-hash 14033 df-struct 16836 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-mulr 16964 df-sca 16966 df-vsca 16967 df-ip 16968 df-tset 16969 df-ple 16970 df-ds 16972 df-hom 16974 df-cco 16975 df-0g 17140 df-gsum 17141 df-prds 17146 df-pws 17148 df-mre 17283 df-mrc 17284 df-acs 17286 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-mhm 18418 df-submnd 18419 df-grp 18568 df-minusg 18569 df-sbg 18570 df-mulg 18689 df-subg 18740 df-ghm 18820 df-cntz 18911 df-cmn 19376 df-abl 19377 df-mgp 19709 df-ur 19726 df-srg 19730 df-ring 19773 df-cring 19774 df-rnghom 19947 df-subrg 20010 df-lmod 20113 df-lss 20182 df-lsp 20222 df-assa 21048 df-asp 21049 df-ascl 21050 df-psr 21100 df-mvr 21101 df-mpl 21102 df-evls 21270 df-evl 21271 |
This theorem is referenced by: (None) |
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