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Mirrors > Home > MPE Home > Th. List > evls1scasrng | Structured version Visualization version GIF version |
Description: The evaluation of a scalar of a subring yields the same result as evaluated as a scalar over the ring itself. (Contributed by AV, 13-Sep-2019.) |
Ref | Expression |
---|---|
evls1scasrng.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
evls1scasrng.o | ⊢ 𝑂 = (eval1‘𝑆) |
evls1scasrng.w | ⊢ 𝑊 = (Poly1‘𝑈) |
evls1scasrng.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evls1scasrng.p | ⊢ 𝑃 = (Poly1‘𝑆) |
evls1scasrng.b | ⊢ 𝐵 = (Base‘𝑆) |
evls1scasrng.a | ⊢ 𝐴 = (algSc‘𝑊) |
evls1scasrng.c | ⊢ 𝐶 = (algSc‘𝑃) |
evls1scasrng.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evls1scasrng.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evls1scasrng.x | ⊢ (𝜑 → 𝑋 ∈ 𝑅) |
Ref | Expression |
---|---|
evls1scasrng | ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1scasrng.c | . . . . . 6 ⊢ 𝐶 = (algSc‘𝑃) | |
2 | evls1scasrng.p | . . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑆) | |
3 | evls1scasrng.s | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
4 | evls1scasrng.b | . . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑆) | |
5 | 4 | ressid 16558 | . . . . . . . . . . 11 ⊢ (𝑆 ∈ CRing → (𝑆 ↾s 𝐵) = 𝑆) |
6 | 5 | eqcomd 2827 | . . . . . . . . . 10 ⊢ (𝑆 ∈ CRing → 𝑆 = (𝑆 ↾s 𝐵)) |
7 | 3, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (𝑆 ↾s 𝐵)) |
8 | 7 | fveq2d 6673 | . . . . . . . 8 ⊢ (𝜑 → (Poly1‘𝑆) = (Poly1‘(𝑆 ↾s 𝐵))) |
9 | 2, 8 | syl5eq 2868 | . . . . . . 7 ⊢ (𝜑 → 𝑃 = (Poly1‘(𝑆 ↾s 𝐵))) |
10 | 9 | fveq2d 6673 | . . . . . 6 ⊢ (𝜑 → (algSc‘𝑃) = (algSc‘(Poly1‘(𝑆 ↾s 𝐵)))) |
11 | 1, 10 | syl5eq 2868 | . . . . 5 ⊢ (𝜑 → 𝐶 = (algSc‘(Poly1‘(𝑆 ↾s 𝐵)))) |
12 | 11 | fveq1d 6671 | . . . 4 ⊢ (𝜑 → (𝐶‘𝑋) = ((algSc‘(Poly1‘(𝑆 ↾s 𝐵)))‘𝑋)) |
13 | 12 | fveq2d 6673 | . . 3 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘(𝐶‘𝑋)) = ((𝑆 evalSub1 𝐵)‘((algSc‘(Poly1‘(𝑆 ↾s 𝐵)))‘𝑋))) |
14 | eqid 2821 | . . . 4 ⊢ (𝑆 evalSub1 𝐵) = (𝑆 evalSub1 𝐵) | |
15 | eqid 2821 | . . . 4 ⊢ (Poly1‘(𝑆 ↾s 𝐵)) = (Poly1‘(𝑆 ↾s 𝐵)) | |
16 | eqid 2821 | . . . 4 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
17 | eqid 2821 | . . . 4 ⊢ (algSc‘(Poly1‘(𝑆 ↾s 𝐵))) = (algSc‘(Poly1‘(𝑆 ↾s 𝐵))) | |
18 | crngring 19307 | . . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
19 | 4 | subrgid 19536 | . . . . 5 ⊢ (𝑆 ∈ Ring → 𝐵 ∈ (SubRing‘𝑆)) |
20 | 3, 18, 19 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑆)) |
21 | evls1scasrng.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
22 | 4 | subrgss 19535 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵) |
23 | 21, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ⊆ 𝐵) |
24 | evls1scasrng.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑅) | |
25 | 23, 24 | sseldd 3967 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
26 | 14, 15, 16, 4, 17, 3, 20, 25 | evls1sca 20485 | . . 3 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘((algSc‘(Poly1‘(𝑆 ↾s 𝐵)))‘𝑋)) = (𝐵 × {𝑋})) |
27 | 13, 26 | eqtrd 2856 | . 2 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘(𝐶‘𝑋)) = (𝐵 × {𝑋})) |
28 | evls1scasrng.o | . . . . 5 ⊢ 𝑂 = (eval1‘𝑆) | |
29 | 28, 4 | evl1fval1 20493 | . . . 4 ⊢ 𝑂 = (𝑆 evalSub1 𝐵) |
30 | 29 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑂 = (𝑆 evalSub1 𝐵)) |
31 | 30 | fveq1d 6671 | . 2 ⊢ (𝜑 → (𝑂‘(𝐶‘𝑋)) = ((𝑆 evalSub1 𝐵)‘(𝐶‘𝑋))) |
32 | evls1scasrng.q | . . 3 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
33 | evls1scasrng.w | . . 3 ⊢ 𝑊 = (Poly1‘𝑈) | |
34 | evls1scasrng.u | . . 3 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
35 | evls1scasrng.a | . . 3 ⊢ 𝐴 = (algSc‘𝑊) | |
36 | 32, 33, 34, 4, 35, 3, 21, 24 | evls1sca 20485 | . 2 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝐵 × {𝑋})) |
37 | 27, 31, 36 | 3eqtr4rd 2867 | 1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 {csn 4566 × cxp 5552 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 ↾s cress 16483 Ringcrg 19296 CRingccrg 19297 SubRingcsubrg 19530 algSccascl 20083 Poly1cpl1 20344 evalSub1 ces1 20475 eval1ce1 20476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-ofr 7409 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-fzo 13033 df-seq 13369 df-hash 13690 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-hom 16588 df-cco 16589 df-0g 16714 df-gsum 16715 df-prds 16720 df-pws 16722 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-cntz 18446 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-srg 19255 df-ring 19298 df-cring 19299 df-rnghom 19466 df-subrg 19532 df-lmod 19635 df-lss 19703 df-lsp 19743 df-assa 20084 df-asp 20085 df-ascl 20086 df-psr 20135 df-mvr 20136 df-mpl 20137 df-opsr 20139 df-evls 20285 df-evl 20286 df-psr1 20347 df-ply1 20349 df-evls1 20477 df-evl1 20478 |
This theorem is referenced by: (None) |
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