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Mirrors > Home > MPE Home > Th. List > evls1var | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation for subrings maps the variable to the identity function. (Contributed by AV, 13-Sep-2019.) |
Ref | Expression |
---|---|
evls1var.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
evls1var.x | ⊢ 𝑋 = (var1‘𝑈) |
evls1var.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evls1var.b | ⊢ 𝐵 = (Base‘𝑆) |
evls1var.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evls1var.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
Ref | Expression |
---|---|
evls1var | ⊢ (𝜑 → (𝑄‘𝑋) = ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1var.x | . . . 4 ⊢ 𝑋 = (var1‘𝑈) | |
2 | eqid 2739 | . . . . 5 ⊢ (var1‘𝑆) = (var1‘𝑆) | |
3 | evls1var.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
4 | evls1var.u | . . . . 5 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
5 | 2, 3, 4 | subrgvr1 21413 | . . . 4 ⊢ (𝜑 → (var1‘𝑆) = (var1‘𝑈)) |
6 | 1, 5 | eqtr4id 2798 | . . 3 ⊢ (𝜑 → 𝑋 = (var1‘𝑆)) |
7 | 6 | fveq2d 6772 | . 2 ⊢ (𝜑 → (𝑄‘𝑋) = (𝑄‘(var1‘𝑆))) |
8 | eqid 2739 | . . . . . 6 ⊢ ((1o evalSub 𝑆)‘𝑅) = ((1o evalSub 𝑆)‘𝑅) | |
9 | eqid 2739 | . . . . . 6 ⊢ (1o eval 𝑆) = (1o eval 𝑆) | |
10 | eqid 2739 | . . . . . 6 ⊢ (1o mVar 𝑈) = (1o mVar 𝑈) | |
11 | evls1var.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑆) | |
12 | 1on 8288 | . . . . . . 7 ⊢ 1o ∈ On | |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1o ∈ On) |
14 | evls1var.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
15 | 0lt1o 8310 | . . . . . . 7 ⊢ ∅ ∈ 1o | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∅ ∈ 1o) |
17 | 8, 9, 10, 4, 11, 13, 14, 3, 16 | evlsvarsrng 21290 | . . . . 5 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘((1o mVar 𝑈)‘∅)) = ((1o eval 𝑆)‘((1o mVar 𝑈)‘∅))) |
18 | 2 | vr1val 21344 | . . . . . . 7 ⊢ (var1‘𝑆) = ((1o mVar 𝑆)‘∅) |
19 | eqid 2739 | . . . . . . . . 9 ⊢ (1o mVar 𝑆) = (1o mVar 𝑆) | |
20 | 19, 13, 3, 4 | subrgmvr 21215 | . . . . . . . 8 ⊢ (𝜑 → (1o mVar 𝑆) = (1o mVar 𝑈)) |
21 | 20 | fveq1d 6770 | . . . . . . 7 ⊢ (𝜑 → ((1o mVar 𝑆)‘∅) = ((1o mVar 𝑈)‘∅)) |
22 | 18, 21 | eqtrid 2791 | . . . . . 6 ⊢ (𝜑 → (var1‘𝑆) = ((1o mVar 𝑈)‘∅)) |
23 | 22 | fveq2d 6772 | . . . . 5 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) = (((1o evalSub 𝑆)‘𝑅)‘((1o mVar 𝑈)‘∅))) |
24 | 22 | fveq2d 6772 | . . . . 5 ⊢ (𝜑 → ((1o eval 𝑆)‘(var1‘𝑆)) = ((1o eval 𝑆)‘((1o mVar 𝑈)‘∅))) |
25 | 17, 23, 24 | 3eqtr4d 2789 | . . . 4 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) = ((1o eval 𝑆)‘(var1‘𝑆))) |
26 | 25 | coeq1d 5767 | . . 3 ⊢ (𝜑 → ((((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (((1o eval 𝑆)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
27 | eqid 2739 | . . . . 5 ⊢ (Poly1‘𝑈) = (Poly1‘𝑈) | |
28 | eqid 2739 | . . . . . . 7 ⊢ (Poly1‘(𝑆 ↾s 𝑅)) = (Poly1‘(𝑆 ↾s 𝑅)) | |
29 | eqid 2739 | . . . . . . 7 ⊢ (PwSer1‘(𝑆 ↾s 𝑅)) = (PwSer1‘(𝑆 ↾s 𝑅)) | |
30 | 4 | fveq2i 6771 | . . . . . . . 8 ⊢ (Poly1‘𝑈) = (Poly1‘(𝑆 ↾s 𝑅)) |
31 | 30 | fveq2i 6771 | . . . . . . 7 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(Poly1‘(𝑆 ↾s 𝑅))) |
32 | 28, 29, 31 | ply1bas 21347 | . . . . . 6 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(1o mPoly (𝑆 ↾s 𝑅))) |
33 | 32 | eqcomi 2748 | . . . . 5 ⊢ (Base‘(1o mPoly (𝑆 ↾s 𝑅))) = (Base‘(Poly1‘𝑈)) |
34 | 2, 3, 4, 27, 33 | subrgvr1cl 21414 | . . . 4 ⊢ (𝜑 → (var1‘𝑆) ∈ (Base‘(1o mPoly (𝑆 ↾s 𝑅)))) |
35 | evls1var.q | . . . . 5 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
36 | eqid 2739 | . . . . 5 ⊢ (1o evalSub 𝑆) = (1o evalSub 𝑆) | |
37 | eqid 2739 | . . . . 5 ⊢ (1o mPoly (𝑆 ↾s 𝑅)) = (1o mPoly (𝑆 ↾s 𝑅)) | |
38 | eqid 2739 | . . . . 5 ⊢ (Base‘(1o mPoly (𝑆 ↾s 𝑅))) = (Base‘(1o mPoly (𝑆 ↾s 𝑅))) | |
39 | 35, 36, 11, 37, 38 | evls1val 21467 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆) ∧ (var1‘𝑆) ∈ (Base‘(1o mPoly (𝑆 ↾s 𝑅)))) → (𝑄‘(var1‘𝑆)) = ((((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
40 | 14, 3, 34, 39 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (𝑄‘(var1‘𝑆)) = ((((1o evalSub 𝑆)‘𝑅)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
41 | crngring 19776 | . . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
42 | eqid 2739 | . . . . . 6 ⊢ (Poly1‘𝑆) = (Poly1‘𝑆) | |
43 | eqid 2739 | . . . . . . . 8 ⊢ (PwSer1‘𝑆) = (PwSer1‘𝑆) | |
44 | eqid 2739 | . . . . . . . 8 ⊢ (Base‘(Poly1‘𝑆)) = (Base‘(Poly1‘𝑆)) | |
45 | 42, 43, 44 | ply1bas 21347 | . . . . . . 7 ⊢ (Base‘(Poly1‘𝑆)) = (Base‘(1o mPoly 𝑆)) |
46 | 45 | eqcomi 2748 | . . . . . 6 ⊢ (Base‘(1o mPoly 𝑆)) = (Base‘(Poly1‘𝑆)) |
47 | 2, 42, 46 | vr1cl 21369 | . . . . 5 ⊢ (𝑆 ∈ Ring → (var1‘𝑆) ∈ (Base‘(1o mPoly 𝑆))) |
48 | 14, 41, 47 | 3syl 18 | . . . 4 ⊢ (𝜑 → (var1‘𝑆) ∈ (Base‘(1o mPoly 𝑆))) |
49 | eqid 2739 | . . . . 5 ⊢ (eval1‘𝑆) = (eval1‘𝑆) | |
50 | eqid 2739 | . . . . 5 ⊢ (1o mPoly 𝑆) = (1o mPoly 𝑆) | |
51 | eqid 2739 | . . . . 5 ⊢ (Base‘(1o mPoly 𝑆)) = (Base‘(1o mPoly 𝑆)) | |
52 | 49, 9, 11, 50, 51 | evl1val 21476 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ (var1‘𝑆) ∈ (Base‘(1o mPoly 𝑆))) → ((eval1‘𝑆)‘(var1‘𝑆)) = (((1o eval 𝑆)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
53 | 14, 48, 52 | syl2anc 583 | . . 3 ⊢ (𝜑 → ((eval1‘𝑆)‘(var1‘𝑆)) = (((1o eval 𝑆)‘(var1‘𝑆)) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) |
54 | 26, 40, 53 | 3eqtr4d 2789 | . 2 ⊢ (𝜑 → (𝑄‘(var1‘𝑆)) = ((eval1‘𝑆)‘(var1‘𝑆))) |
55 | 49, 2, 11 | evl1var 21483 | . . 3 ⊢ (𝑆 ∈ CRing → ((eval1‘𝑆)‘(var1‘𝑆)) = ( I ↾ 𝐵)) |
56 | 14, 55 | syl 17 | . 2 ⊢ (𝜑 → ((eval1‘𝑆)‘(var1‘𝑆)) = ( I ↾ 𝐵)) |
57 | 7, 54, 56 | 3eqtrd 2783 | 1 ⊢ (𝜑 → (𝑄‘𝑋) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2109 ∅c0 4261 {csn 4566 ↦ cmpt 5161 I cid 5487 × cxp 5586 ↾ cres 5590 ∘ ccom 5592 Oncon0 6263 ‘cfv 6430 (class class class)co 7268 1oc1o 8274 Basecbs 16893 ↾s cress 16922 Ringcrg 19764 CRingccrg 19765 SubRingcsubrg 20001 mVar cmvr 21089 mPoly cmpl 21090 evalSub ces 21261 eval cevl 21262 PwSer1cps1 21327 var1cv1 21328 Poly1cpl1 21329 evalSub1 ces1 21460 eval1ce1 21461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-ofr 7525 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-pm 8592 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-sup 9162 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-fz 13222 df-fzo 13365 df-seq 13703 df-hash 14026 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-hom 16967 df-cco 16968 df-0g 17133 df-gsum 17134 df-prds 17139 df-pws 17141 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-mhm 18411 df-submnd 18412 df-grp 18561 df-minusg 18562 df-sbg 18563 df-mulg 18682 df-subg 18733 df-ghm 18813 df-cntz 18904 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-srg 19723 df-ring 19766 df-cring 19767 df-rnghom 19940 df-subrg 20003 df-lmod 20106 df-lss 20175 df-lsp 20215 df-assa 21041 df-asp 21042 df-ascl 21043 df-psr 21093 df-mvr 21094 df-mpl 21095 df-opsr 21097 df-evls 21263 df-evl 21264 df-psr1 21332 df-vr1 21333 df-ply1 21334 df-evls1 21462 df-evl1 21463 |
This theorem is referenced by: evls1varsrng 21487 |
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