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Mirrors > Home > MPE Home > Th. List > Mathboxes > selvval2lemn | Structured version Visualization version GIF version |
Description: A lemma to illustrate the purpose of selvval2lem3 40607 and the value of 𝑄. Will be renamed in the future when this section is moved to main. (Contributed by SN, 5-Nov-2023.) |
Ref | Expression |
---|---|
selvval2lemn.u | ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) |
selvval2lemn.t | ⊢ 𝑇 = (𝐽 mPoly 𝑈) |
selvval2lemn.c | ⊢ 𝐶 = (algSc‘𝑇) |
selvval2lemn.d | ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) |
selvval2lemn.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑇)‘ran 𝐷) |
selvval2lemn.w | ⊢ 𝑊 = (𝐼 mPoly 𝑆) |
selvval2lemn.s | ⊢ 𝑆 = (𝑇 ↾s ran 𝐷) |
selvval2lemn.x | ⊢ 𝑋 = (𝑇 ↑s (𝐵 ↑m 𝐼)) |
selvval2lemn.b | ⊢ 𝐵 = (Base‘𝑇) |
selvval2lemn.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
selvval2lemn.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
selvval2lemn.j | ⊢ (𝜑 → 𝐽 ⊆ 𝐼) |
Ref | Expression |
---|---|
selvval2lemn | ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | selvval2lemn.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
2 | selvval2lemn.j | . . . 4 ⊢ (𝜑 → 𝐽 ⊆ 𝐼) | |
3 | 1, 2 | ssexd 5279 | . . 3 ⊢ (𝜑 → 𝐽 ∈ V) |
4 | 1 | difexd 5284 | . . . 4 ⊢ (𝜑 → (𝐼 ∖ 𝐽) ∈ V) |
5 | selvval2lemn.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
6 | selvval2lemn.u | . . . . 5 ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) | |
7 | 6 | mplcrng 21372 | . . . 4 ⊢ (((𝐼 ∖ 𝐽) ∈ V ∧ 𝑅 ∈ CRing) → 𝑈 ∈ CRing) |
8 | 4, 5, 7 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑈 ∈ CRing) |
9 | selvval2lemn.t | . . . 4 ⊢ 𝑇 = (𝐽 mPoly 𝑈) | |
10 | 9 | mplcrng 21372 | . . 3 ⊢ ((𝐽 ∈ V ∧ 𝑈 ∈ CRing) → 𝑇 ∈ CRing) |
11 | 3, 8, 10 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝑇 ∈ CRing) |
12 | selvval2lemn.c | . . 3 ⊢ 𝐶 = (algSc‘𝑇) | |
13 | selvval2lemn.d | . . 3 ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) | |
14 | 6, 9, 12, 13, 4, 3, 5 | selvval2lem3 40607 | . 2 ⊢ (𝜑 → ran 𝐷 ∈ (SubRing‘𝑇)) |
15 | selvval2lemn.q | . . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑇)‘ran 𝐷) | |
16 | selvval2lemn.w | . . 3 ⊢ 𝑊 = (𝐼 mPoly 𝑆) | |
17 | selvval2lemn.s | . . 3 ⊢ 𝑆 = (𝑇 ↾s ran 𝐷) | |
18 | selvval2lemn.x | . . 3 ⊢ 𝑋 = (𝑇 ↑s (𝐵 ↑m 𝐼)) | |
19 | selvval2lemn.b | . . 3 ⊢ 𝐵 = (Base‘𝑇) | |
20 | 15, 16, 17, 18, 19 | evlsrhm 21444 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑇 ∈ CRing ∧ ran 𝐷 ∈ (SubRing‘𝑇)) → 𝑄 ∈ (𝑊 RingHom 𝑋)) |
21 | 1, 11, 14, 20 | syl3anc 1371 | 1 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∖ cdif 3905 ⊆ wss 3908 ran crn 5632 ∘ ccom 5635 ‘cfv 6493 (class class class)co 7351 ↑m cmap 8723 Basecbs 17037 ↾s cress 17066 ↑s cpws 17282 CRingccrg 19913 RingHom crh 20090 SubRingcsubrg 20165 algSccascl 21205 mPoly cmpl 21255 evalSub ces 21426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-ofr 7610 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-pm 8726 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-sup 9336 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-fzo 13522 df-seq 13861 df-hash 14185 df-struct 16973 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-plusg 17100 df-mulr 17101 df-sca 17103 df-vsca 17104 df-ip 17105 df-tset 17106 df-ple 17107 df-ds 17109 df-hom 17111 df-cco 17112 df-0g 17277 df-gsum 17278 df-prds 17283 df-pws 17285 df-mre 17420 df-mrc 17421 df-acs 17423 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-mhm 18555 df-submnd 18556 df-grp 18705 df-minusg 18706 df-sbg 18707 df-mulg 18826 df-subg 18878 df-ghm 18959 df-cntz 19050 df-cmn 19517 df-abl 19518 df-mgp 19850 df-ur 19867 df-srg 19871 df-ring 19914 df-cring 19915 df-rnghom 20093 df-subrg 20167 df-lmod 20271 df-lss 20340 df-lsp 20380 df-assa 21206 df-asp 21207 df-ascl 21208 df-psr 21258 df-mvr 21259 df-mpl 21260 df-evls 21428 |
This theorem is referenced by: selvcl 40611 |
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