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| Mirrors > Home > MPE Home > Th. List > Mathboxes > selvcllem4 | Structured version Visualization version GIF version | ||
| Description: The fourth argument passed to evalSub is in the domain (a polynomial in (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))). (Contributed by SN, 5-Nov-2023.) | 
| Ref | Expression | 
|---|---|
| selvcllem4.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) | 
| selvcllem4.b | ⊢ 𝐵 = (Base‘𝑃) | 
| selvcllem4.u | ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) | 
| selvcllem4.t | ⊢ 𝑇 = (𝐽 mPoly 𝑈) | 
| selvcllem4.c | ⊢ 𝐶 = (algSc‘𝑇) | 
| selvcllem4.d | ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) | 
| selvcllem4.s | ⊢ 𝑆 = (𝑇 ↾s ran 𝐷) | 
| selvcllem4.w | ⊢ 𝑊 = (𝐼 mPoly 𝑆) | 
| selvcllem4.x | ⊢ 𝑋 = (Base‘𝑊) | 
| selvcllem4.r | ⊢ (𝜑 → 𝑅 ∈ CRing) | 
| selvcllem4.j | ⊢ (𝜑 → 𝐽 ⊆ 𝐼) | 
| selvcllem4.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| selvcllem4 | ⊢ (𝜑 → (𝐷 ∘ 𝐹) ∈ 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | selvcllem4.p | . 2 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
| 2 | selvcllem4.w | . 2 ⊢ 𝑊 = (𝐼 mPoly 𝑆) | |
| 3 | selvcllem4.b | . 2 ⊢ 𝐵 = (Base‘𝑃) | |
| 4 | selvcllem4.x | . 2 ⊢ 𝑋 = (Base‘𝑊) | |
| 5 | selvcllem4.u | . . . . 5 ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) | |
| 6 | selvcllem4.t | . . . . 5 ⊢ 𝑇 = (𝐽 mPoly 𝑈) | |
| 7 | selvcllem4.c | . . . . 5 ⊢ 𝐶 = (algSc‘𝑇) | |
| 8 | selvcllem4.d | . . . . 5 ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) | |
| 9 | selvcllem4.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
| 10 | 1, 3 | mplrcl 22015 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐵 → 𝐼 ∈ V) | 
| 11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) | 
| 12 | 11 | difexd 5330 | . . . . 5 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V) | 
| 13 | selvcllem4.j | . . . . . 6 ⊢ (𝜑 → 𝐽 ⊆ 𝐼) | |
| 14 | 11, 13 | ssexd 5323 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ V) | 
| 15 | selvcllem4.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
| 16 | 5, 6, 7, 8, 12, 14, 15 | selvcllem2 42593 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝑅 RingHom 𝑇)) | 
| 17 | 5, 6, 7, 8, 12, 14, 15 | selvcllem3 42594 | . . . . 5 ⊢ (𝜑 → ran 𝐷 ∈ (SubRing‘𝑇)) | 
| 18 | ssidd 4006 | . . . . 5 ⊢ (𝜑 → ran 𝐷 ⊆ ran 𝐷) | |
| 19 | selvcllem4.s | . . . . . 6 ⊢ 𝑆 = (𝑇 ↾s ran 𝐷) | |
| 20 | 19 | resrhm2b 20603 | . . . . 5 ⊢ ((ran 𝐷 ∈ (SubRing‘𝑇) ∧ ran 𝐷 ⊆ ran 𝐷) → (𝐷 ∈ (𝑅 RingHom 𝑇) ↔ 𝐷 ∈ (𝑅 RingHom 𝑆))) | 
| 21 | 17, 18, 20 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐷 ∈ (𝑅 RingHom 𝑇) ↔ 𝐷 ∈ (𝑅 RingHom 𝑆))) | 
| 22 | 16, 21 | mpbid 232 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (𝑅 RingHom 𝑆)) | 
| 23 | rhmghm 20485 | . . 3 ⊢ (𝐷 ∈ (𝑅 RingHom 𝑆) → 𝐷 ∈ (𝑅 GrpHom 𝑆)) | |
| 24 | ghmmhm 19245 | . . 3 ⊢ (𝐷 ∈ (𝑅 GrpHom 𝑆) → 𝐷 ∈ (𝑅 MndHom 𝑆)) | |
| 25 | 22, 23, 24 | 3syl 18 | . 2 ⊢ (𝜑 → 𝐷 ∈ (𝑅 MndHom 𝑆)) | 
| 26 | 1, 2, 3, 4, 25, 9 | mhmcompl 22385 | 1 ⊢ (𝜑 → (𝐷 ∘ 𝐹) ∈ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∖ cdif 3947 ⊆ wss 3950 ran crn 5685 ∘ ccom 5688 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 ↾s cress 17275 MndHom cmhm 18795 GrpHom cghm 19231 CRingccrg 20232 RingHom crh 20470 SubRingcsubrg 20570 algSccascl 21873 mPoly cmpl 21927 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-lmod 20861 df-lss 20931 df-assa 21874 df-ascl 21876 df-psr 21930 df-mpl 21932 | 
| This theorem is referenced by: selvcl 42598 selvval2 42599 | 
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