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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 16675 | . . . . . . . . . . 11 ⊢ (𝑆 ∈ CRing → (𝑆 ↾s 𝐵) = 𝑆) |
6 | 5 | eqcomd 2745 | . . . . . . . . . 10 ⊢ (𝑆 ∈ CRing → 𝑆 = (𝑆 ↾s 𝐵)) |
7 | 3, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (𝑆 ↾s 𝐵)) |
8 | 7 | fveq2d 6691 | . . . . . . . 8 ⊢ (𝜑 → (Poly1‘𝑆) = (Poly1‘(𝑆 ↾s 𝐵))) |
9 | 2, 8 | syl5eq 2786 | . . . . . . 7 ⊢ (𝜑 → 𝑃 = (Poly1‘(𝑆 ↾s 𝐵))) |
10 | 9 | fveq2d 6691 | . . . . . 6 ⊢ (𝜑 → (algSc‘𝑃) = (algSc‘(Poly1‘(𝑆 ↾s 𝐵)))) |
11 | 1, 10 | syl5eq 2786 | . . . . 5 ⊢ (𝜑 → 𝐶 = (algSc‘(Poly1‘(𝑆 ↾s 𝐵)))) |
12 | 11 | fveq1d 6689 | . . . 4 ⊢ (𝜑 → (𝐶‘𝑋) = ((algSc‘(Poly1‘(𝑆 ↾s 𝐵)))‘𝑋)) |
13 | 12 | fveq2d 6691 | . . 3 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘(𝐶‘𝑋)) = ((𝑆 evalSub1 𝐵)‘((algSc‘(Poly1‘(𝑆 ↾s 𝐵)))‘𝑋))) |
14 | eqid 2739 | . . . 4 ⊢ (𝑆 evalSub1 𝐵) = (𝑆 evalSub1 𝐵) | |
15 | eqid 2739 | . . . 4 ⊢ (Poly1‘(𝑆 ↾s 𝐵)) = (Poly1‘(𝑆 ↾s 𝐵)) | |
16 | eqid 2739 | . . . 4 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
17 | eqid 2739 | . . . 4 ⊢ (algSc‘(Poly1‘(𝑆 ↾s 𝐵))) = (algSc‘(Poly1‘(𝑆 ↾s 𝐵))) | |
18 | crngring 19441 | . . . . 5 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
19 | 4 | subrgid 19669 | . . . . 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 19668 | . . . . . 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 3888 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
26 | 14, 15, 16, 4, 17, 3, 20, 25 | evls1sca 21106 | . . 3 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘((algSc‘(Poly1‘(𝑆 ↾s 𝐵)))‘𝑋)) = (𝐵 × {𝑋})) |
27 | 13, 26 | eqtrd 2774 | . 2 ⊢ (𝜑 → ((𝑆 evalSub1 𝐵)‘(𝐶‘𝑋)) = (𝐵 × {𝑋})) |
28 | evls1scasrng.o | . . . . 5 ⊢ 𝑂 = (eval1‘𝑆) | |
29 | 28, 4 | evl1fval1 21114 | . . . 4 ⊢ 𝑂 = (𝑆 evalSub1 𝐵) |
30 | 29 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑂 = (𝑆 evalSub1 𝐵)) |
31 | 30 | fveq1d 6689 | . 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 21106 | . 2 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝐵 × {𝑋})) |
37 | 27, 31, 36 | 3eqtr4rd 2785 | 1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝑂‘(𝐶‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⊆ wss 3853 {csn 4526 × cxp 5533 ‘cfv 6350 (class class class)co 7183 Basecbs 16599 ↾s cress 16600 Ringcrg 19429 CRingccrg 19430 SubRingcsubrg 19663 algSccascl 20681 Poly1cpl1 20965 evalSub1 ces1 21096 eval1ce1 21097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-of 7438 df-ofr 7439 df-om 7613 df-1st 7727 df-2nd 7728 df-supp 7870 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-pm 8453 df-ixp 8521 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-fsupp 8920 df-sup 8992 df-oi 9060 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-z 12076 df-dec 12193 df-uz 12338 df-fz 12995 df-fzo 13138 df-seq 13474 df-hash 13796 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-sca 16697 df-vsca 16698 df-ip 16699 df-tset 16700 df-ple 16701 df-ds 16703 df-hom 16705 df-cco 16706 df-0g 16831 df-gsum 16832 df-prds 16837 df-pws 16839 df-mre 16973 df-mrc 16974 df-acs 16976 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-mhm 18085 df-submnd 18086 df-grp 18235 df-minusg 18236 df-sbg 18237 df-mulg 18356 df-subg 18407 df-ghm 18487 df-cntz 18578 df-cmn 19039 df-abl 19040 df-mgp 19372 df-ur 19384 df-srg 19388 df-ring 19431 df-cring 19432 df-rnghom 19602 df-subrg 19665 df-lmod 19768 df-lss 19836 df-lsp 19876 df-assa 20682 df-asp 20683 df-ascl 20684 df-psr 20735 df-mvr 20736 df-mpl 20737 df-opsr 20739 df-evls 20899 df-evl 20900 df-psr1 20968 df-ply1 20970 df-evls1 21098 df-evl1 21099 |
This theorem is referenced by: (None) |
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