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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ply1annnr | Structured version Visualization version GIF version |
Description: The set 𝑄 of polynomials annihilating an element 𝐴 is not the whole polynomial ring. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
Ref | Expression |
---|---|
ply1annidl.o | ⊢ 𝑂 = (𝑅 evalSub1 𝑆) |
ply1annidl.p | ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆)) |
ply1annidl.b | ⊢ 𝐵 = (Base‘𝑅) |
ply1annidl.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
ply1annidl.s | ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) |
ply1annidl.a | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
ply1annidl.0 | ⊢ 0 = (0g‘𝑅) |
ply1annidl.q | ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } |
ply1annnr.u | ⊢ 𝑈 = (Base‘𝑃) |
ply1annnr.1 | ⊢ (𝜑 → 𝑅 ∈ NzRing) |
Ref | Expression |
---|---|
ply1annnr | ⊢ (𝜑 → 𝑄 ≠ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1annidl.q | . . 3 ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) |
3 | ply1annidl.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
4 | 3 | crngringd 20144 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring) |
5 | ply1annidl.s | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) | |
6 | eqid 2731 | . . . . . . . . . 10 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | 6 | subrg1cl 20474 | . . . . . . . . 9 ⊢ (𝑆 ∈ (SubRing‘𝑅) → (1r‘𝑅) ∈ 𝑆) |
8 | 5, 7 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝑆) |
9 | ply1annidl.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅) | |
10 | 9 | subrgss 20466 | . . . . . . . . 9 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵) |
11 | 5, 10 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
12 | eqid 2731 | . . . . . . . . 9 ⊢ (𝑅 ↾s 𝑆) = (𝑅 ↾s 𝑆) | |
13 | 12, 9, 6 | ress1r 32668 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ 𝑆 ∧ 𝑆 ⊆ 𝐵) → (1r‘𝑅) = (1r‘(𝑅 ↾s 𝑆))) |
14 | 4, 8, 11, 13 | syl3anc 1370 | . . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) = (1r‘(𝑅 ↾s 𝑆))) |
15 | 14 | fveq2d 6895 | . . . . . 6 ⊢ (𝜑 → ((algSc‘𝑃)‘(1r‘𝑅)) = ((algSc‘𝑃)‘(1r‘(𝑅 ↾s 𝑆)))) |
16 | ply1annidl.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘(𝑅 ↾s 𝑆)) | |
17 | eqid 2731 | . . . . . . 7 ⊢ (algSc‘𝑃) = (algSc‘𝑃) | |
18 | eqid 2731 | . . . . . . 7 ⊢ (1r‘(𝑅 ↾s 𝑆)) = (1r‘(𝑅 ↾s 𝑆)) | |
19 | eqid 2731 | . . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
20 | 12 | subrgring 20468 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝑆) ∈ Ring) |
21 | 5, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ Ring) |
22 | 16, 17, 18, 19, 21 | ply1ascl1 32942 | . . . . . 6 ⊢ (𝜑 → ((algSc‘𝑃)‘(1r‘(𝑅 ↾s 𝑆))) = (1r‘𝑃)) |
23 | 15, 22 | eqtrd 2771 | . . . . 5 ⊢ (𝜑 → ((algSc‘𝑃)‘(1r‘𝑅)) = (1r‘𝑃)) |
24 | 16 | ply1ring 22003 | . . . . . 6 ⊢ ((𝑅 ↾s 𝑆) ∈ Ring → 𝑃 ∈ Ring) |
25 | ply1annnr.u | . . . . . . 7 ⊢ 𝑈 = (Base‘𝑃) | |
26 | 25, 19 | ringidcl 20158 | . . . . . 6 ⊢ (𝑃 ∈ Ring → (1r‘𝑃) ∈ 𝑈) |
27 | 21, 24, 26 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (1r‘𝑃) ∈ 𝑈) |
28 | 23, 27 | eqeltrd 2832 | . . . 4 ⊢ (𝜑 → ((algSc‘𝑃)‘(1r‘𝑅)) ∈ 𝑈) |
29 | ply1annidl.o | . . . . . . . 8 ⊢ 𝑂 = (𝑅 evalSub1 𝑆) | |
30 | ply1annidl.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
31 | 29, 16, 12, 9, 17, 3, 5, 8, 30 | evls1scafv 32932 | . . . . . . 7 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑃)‘(1r‘𝑅)))‘𝐴) = (1r‘𝑅)) |
32 | ply1annnr.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ NzRing) | |
33 | ply1annidl.0 | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
34 | 6, 33 | nzrnz 20410 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ 0 ) |
35 | 32, 34 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (1r‘𝑅) ≠ 0 ) |
36 | 31, 35 | eqnetrd 3007 | . . . . . 6 ⊢ (𝜑 → ((𝑂‘((algSc‘𝑃)‘(1r‘𝑅)))‘𝐴) ≠ 0 ) |
37 | 36 | neneqd 2944 | . . . . 5 ⊢ (𝜑 → ¬ ((𝑂‘((algSc‘𝑃)‘(1r‘𝑅)))‘𝐴) = 0 ) |
38 | fveq2 6891 | . . . . . . . . 9 ⊢ (𝑞 = ((algSc‘𝑃)‘(1r‘𝑅)) → (𝑂‘𝑞) = (𝑂‘((algSc‘𝑃)‘(1r‘𝑅)))) | |
39 | 38 | fveq1d 6893 | . . . . . . . 8 ⊢ (𝑞 = ((algSc‘𝑃)‘(1r‘𝑅)) → ((𝑂‘𝑞)‘𝐴) = ((𝑂‘((algSc‘𝑃)‘(1r‘𝑅)))‘𝐴)) |
40 | 39 | eqeq1d 2733 | . . . . . . 7 ⊢ (𝑞 = ((algSc‘𝑃)‘(1r‘𝑅)) → (((𝑂‘𝑞)‘𝐴) = 0 ↔ ((𝑂‘((algSc‘𝑃)‘(1r‘𝑅)))‘𝐴) = 0 )) |
41 | 40 | elrab 3683 | . . . . . 6 ⊢ (((algSc‘𝑃)‘(1r‘𝑅)) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ↔ (((algSc‘𝑃)‘(1r‘𝑅)) ∈ dom 𝑂 ∧ ((𝑂‘((algSc‘𝑃)‘(1r‘𝑅)))‘𝐴) = 0 )) |
42 | 41 | simprbi 496 | . . . . 5 ⊢ (((algSc‘𝑃)‘(1r‘𝑅)) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } → ((𝑂‘((algSc‘𝑃)‘(1r‘𝑅)))‘𝐴) = 0 ) |
43 | 37, 42 | nsyl 140 | . . . 4 ⊢ (𝜑 → ¬ ((algSc‘𝑃)‘(1r‘𝑅)) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) |
44 | nelne1 3038 | . . . 4 ⊢ ((((algSc‘𝑃)‘(1r‘𝑅)) ∈ 𝑈 ∧ ¬ ((algSc‘𝑃)‘(1r‘𝑅)) ∈ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) → 𝑈 ≠ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) | |
45 | 28, 43, 44 | syl2anc 583 | . . 3 ⊢ (𝜑 → 𝑈 ≠ {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }) |
46 | 45 | necomd 2995 | . 2 ⊢ (𝜑 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ≠ 𝑈) |
47 | 2, 46 | eqnetrd 3007 | 1 ⊢ (𝜑 → 𝑄 ≠ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 {crab 3431 ⊆ wss 3948 dom cdm 5676 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 ↾s cress 17180 0gc0g 17392 1rcur 20079 Ringcrg 20131 CRingccrg 20132 NzRingcnzr 20407 SubRingcsubrg 20461 algSccascl 21630 Poly1cpl1 21933 evalSub1 ces1 22065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-mhm 18708 df-submnd 18709 df-grp 18861 df-minusg 18862 df-sbg 18863 df-mulg 18991 df-subg 19043 df-ghm 19132 df-cntz 19226 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-srg 20085 df-ring 20133 df-cring 20134 df-rhm 20367 df-nzr 20408 df-subrng 20438 df-subrg 20463 df-lmod 20620 df-lss 20691 df-lsp 20731 df-assa 21631 df-asp 21632 df-ascl 21633 df-psr 21685 df-mvr 21686 df-mpl 21687 df-opsr 21689 df-evls 21859 df-psr1 21936 df-ply1 21938 df-evls1 22067 |
This theorem is referenced by: minplyirred 33074 |
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