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Mirrors > Home > MPE Home > Th. List > Mathboxes > selvcllem5 | Structured version Visualization version GIF version |
Description: The fifth argument passed to evalSub is in the domain (a function πΌβΆπΈ). (Contributed by SN, 22-Feb-2024.) |
Ref | Expression |
---|---|
selvcllem5.u | β’ π = ((πΌ β π½) mPoly π ) |
selvcllem5.t | β’ π = (π½ mPoly π) |
selvcllem5.c | β’ πΆ = (algScβπ) |
selvcllem5.e | β’ πΈ = (Baseβπ) |
selvcllem5.f | β’ πΉ = (π₯ β πΌ β¦ if(π₯ β π½, ((π½ mVar π)βπ₯), (πΆβ(((πΌ β π½) mVar π )βπ₯)))) |
selvcllem5.i | β’ (π β πΌ β π) |
selvcllem5.r | β’ (π β π β CRing) |
selvcllem5.j | β’ (π β π½ β πΌ) |
Ref | Expression |
---|---|
selvcllem5 | β’ (π β πΉ β (πΈ βm πΌ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | selvcllem5.t | . . . . . 6 β’ π = (π½ mPoly π) | |
2 | eqid 2726 | . . . . . 6 β’ (π½ mVar π) = (π½ mVar π) | |
3 | selvcllem5.e | . . . . . 6 β’ πΈ = (Baseβπ) | |
4 | selvcllem5.i | . . . . . . . 8 β’ (π β πΌ β π) | |
5 | selvcllem5.j | . . . . . . . 8 β’ (π β π½ β πΌ) | |
6 | 4, 5 | ssexd 5317 | . . . . . . 7 β’ (π β π½ β V) |
7 | 6 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β π½) β π½ β V) |
8 | 4 | difexd 5322 | . . . . . . . 8 β’ (π β (πΌ β π½) β V) |
9 | selvcllem5.r | . . . . . . . . 9 β’ (π β π β CRing) | |
10 | crngring 20150 | . . . . . . . . 9 β’ (π β CRing β π β Ring) | |
11 | 9, 10 | syl 17 | . . . . . . . 8 β’ (π β π β Ring) |
12 | selvcllem5.u | . . . . . . . . 9 β’ π = ((πΌ β π½) mPoly π ) | |
13 | 12 | mplring 21920 | . . . . . . . 8 β’ (((πΌ β π½) β V β§ π β Ring) β π β Ring) |
14 | 8, 11, 13 | syl2anc 583 | . . . . . . 7 β’ (π β π β Ring) |
15 | 14 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β π½) β π β Ring) |
16 | simpr 484 | . . . . . 6 β’ ((π β§ π₯ β π½) β π₯ β π½) | |
17 | 1, 2, 3, 7, 15, 16 | mvrcl 21893 | . . . . 5 β’ ((π β§ π₯ β π½) β ((π½ mVar π)βπ₯) β πΈ) |
18 | 17 | adantlr 712 | . . . 4 β’ (((π β§ π₯ β πΌ) β§ π₯ β π½) β ((π½ mVar π)βπ₯) β πΈ) |
19 | eqid 2726 | . . . . . . 7 β’ (Baseβπ) = (Baseβπ) | |
20 | selvcllem5.c | . . . . . . 7 β’ πΆ = (algScβπ) | |
21 | 1, 3, 19, 20, 6, 14 | mplasclf 21968 | . . . . . 6 β’ (π β πΆ:(Baseβπ)βΆπΈ) |
22 | 21 | ad2antrr 723 | . . . . 5 β’ (((π β§ π₯ β πΌ) β§ Β¬ π₯ β π½) β πΆ:(Baseβπ)βΆπΈ) |
23 | eqid 2726 | . . . . . 6 β’ ((πΌ β π½) mVar π ) = ((πΌ β π½) mVar π ) | |
24 | 8 | ad2antrr 723 | . . . . . 6 β’ (((π β§ π₯ β πΌ) β§ Β¬ π₯ β π½) β (πΌ β π½) β V) |
25 | 11 | ad2antrr 723 | . . . . . 6 β’ (((π β§ π₯ β πΌ) β§ Β¬ π₯ β π½) β π β Ring) |
26 | eldif 3953 | . . . . . . . 8 β’ (π₯ β (πΌ β π½) β (π₯ β πΌ β§ Β¬ π₯ β π½)) | |
27 | 26 | biimpri 227 | . . . . . . 7 β’ ((π₯ β πΌ β§ Β¬ π₯ β π½) β π₯ β (πΌ β π½)) |
28 | 27 | adantll 711 | . . . . . 6 β’ (((π β§ π₯ β πΌ) β§ Β¬ π₯ β π½) β π₯ β (πΌ β π½)) |
29 | 12, 23, 19, 24, 25, 28 | mvrcl 21893 | . . . . 5 β’ (((π β§ π₯ β πΌ) β§ Β¬ π₯ β π½) β (((πΌ β π½) mVar π )βπ₯) β (Baseβπ)) |
30 | 22, 29 | ffvelcdmd 7081 | . . . 4 β’ (((π β§ π₯ β πΌ) β§ Β¬ π₯ β π½) β (πΆβ(((πΌ β π½) mVar π )βπ₯)) β πΈ) |
31 | 18, 30 | ifclda 4558 | . . 3 β’ ((π β§ π₯ β πΌ) β if(π₯ β π½, ((π½ mVar π)βπ₯), (πΆβ(((πΌ β π½) mVar π )βπ₯))) β πΈ) |
32 | selvcllem5.f | . . 3 β’ πΉ = (π₯ β πΌ β¦ if(π₯ β π½, ((π½ mVar π)βπ₯), (πΆβ(((πΌ β π½) mVar π )βπ₯)))) | |
33 | 31, 32 | fmptd 7109 | . 2 β’ (π β πΉ:πΌβΆπΈ) |
34 | fvexd 6900 | . . . 4 β’ (π β (Baseβπ) β V) | |
35 | 3, 34 | eqeltrid 2831 | . . 3 β’ (π β πΈ β V) |
36 | 35, 4 | elmapd 8836 | . 2 β’ (π β (πΉ β (πΈ βm πΌ) β πΉ:πΌβΆπΈ)) |
37 | 33, 36 | mpbird 257 | 1 β’ (π β πΉ β (πΈ βm πΌ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 β cdif 3940 β wss 3943 ifcif 4523 β¦ cmpt 5224 βΆwf 6533 βcfv 6537 (class class class)co 7405 βm cmap 8822 Basecbs 17153 Ringcrg 20138 CRingccrg 20139 algSccascl 21747 mVar cmvr 21799 mPoly cmpl 21800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-subrng 20446 df-subrg 20471 df-lmod 20708 df-lss 20779 df-ascl 21750 df-psr 21803 df-mvr 21804 df-mpl 21805 |
This theorem is referenced by: selvcl 41717 selvadd 41722 selvmul 41723 |
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