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Mirrors > Home > MPE Home > Th. List > ply1nzb | Structured version Visualization version GIF version |
Description: Univariate polynomials are nonzero iff the base is nonzero. Or in contraposition, the univariate polynomials over the zero ring are also zero. (Contributed by Mario Carneiro, 13-Jun-2015.) |
Ref | Expression |
---|---|
ply1domn.p | ⊢ 𝑃 = (Poly1‘𝑅) |
Ref | Expression |
---|---|
ply1nzb | ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ 𝑃 ∈ NzRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1domn.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | 1 | ply1nz 25502 | . 2 ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ NzRing) |
3 | simpl 484 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → 𝑅 ∈ Ring) | |
4 | eqid 2733 | . . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
5 | eqid 2733 | . . . . . . 7 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
6 | 4, 5 | nzrnz 20746 | . . . . . 6 ⊢ (𝑃 ∈ NzRing → (1r‘𝑃) ≠ (0g‘𝑃)) |
7 | 6 | adantl 483 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → (1r‘𝑃) ≠ (0g‘𝑃)) |
8 | ifeq1 4491 | . . . . . . . . 9 ⊢ ((1r‘𝑅) = (0g‘𝑅) → if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅)) = if(𝑦 = (1o × {0}), (0g‘𝑅), (0g‘𝑅))) | |
9 | ifid 4527 | . . . . . . . . 9 ⊢ if(𝑦 = (1o × {0}), (0g‘𝑅), (0g‘𝑅)) = (0g‘𝑅) | |
10 | 8, 9 | eqtrdi 2789 | . . . . . . . 8 ⊢ ((1r‘𝑅) = (0g‘𝑅) → if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅)) |
11 | 10 | ralrimivw 3144 | . . . . . . 7 ⊢ ((1r‘𝑅) = (0g‘𝑅) → ∀𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin}if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅)) |
12 | eqid 2733 | . . . . . . . . . 10 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
13 | eqid 2733 | . . . . . . . . . 10 ⊢ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} = {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} | |
14 | eqid 2733 | . . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
15 | eqid 2733 | . . . . . . . . . 10 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
16 | 12, 1, 4 | ply1mpl1 21644 | . . . . . . . . . 10 ⊢ (1r‘𝑃) = (1r‘(1o mPoly 𝑅)) |
17 | 1on 8425 | . . . . . . . . . . 11 ⊢ 1o ∈ On | |
18 | 17 | a1i 11 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → 1o ∈ On) |
19 | 12, 13, 14, 15, 16, 18, 3 | mpl1 21432 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → (1r‘𝑃) = (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅)))) |
20 | 12, 1, 5 | ply1mpl0 21642 | . . . . . . . . . . 11 ⊢ (0g‘𝑃) = (0g‘(1o mPoly 𝑅)) |
21 | ringgrp 19974 | . . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
22 | 3, 21 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → 𝑅 ∈ Grp) |
23 | 12, 13, 14, 20, 18, 22 | mpl0 21428 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → (0g‘𝑃) = ({𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {(0g‘𝑅)})) |
24 | fconstmpt 5695 | . . . . . . . . . 10 ⊢ ({𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {(0g‘𝑅)}) = (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (0g‘𝑅)) | |
25 | 23, 24 | eqtrdi 2789 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → (0g‘𝑃) = (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (0g‘𝑅))) |
26 | 19, 25 | eqeq12d 2749 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → ((1r‘𝑃) = (0g‘𝑃) ↔ (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅))) = (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (0g‘𝑅)))) |
27 | fvex 6856 | . . . . . . . . . . 11 ⊢ (1r‘𝑅) ∈ V | |
28 | fvex 6856 | . . . . . . . . . . 11 ⊢ (0g‘𝑅) ∈ V | |
29 | 27, 28 | ifex 4537 | . . . . . . . . . 10 ⊢ if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅)) ∈ V |
30 | 29 | rgenw 3065 | . . . . . . . . 9 ⊢ ∀𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin}if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅)) ∈ V |
31 | mpteqb 6968 | . . . . . . . . 9 ⊢ (∀𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin}if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅)) ∈ V → ((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅))) = (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (0g‘𝑅)) ↔ ∀𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin}if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))) | |
32 | 30, 31 | ax-mp 5 | . . . . . . . 8 ⊢ ((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅))) = (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (0g‘𝑅)) ↔ ∀𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin}if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅)) |
33 | 26, 32 | bitrdi 287 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → ((1r‘𝑃) = (0g‘𝑃) ↔ ∀𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 1o) ∣ (◡𝑥 “ ℕ) ∈ Fin}if(𝑦 = (1o × {0}), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅))) |
34 | 11, 33 | syl5ibr 246 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → ((1r‘𝑅) = (0g‘𝑅) → (1r‘𝑃) = (0g‘𝑃))) |
35 | 34 | necon3d 2961 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → ((1r‘𝑃) ≠ (0g‘𝑃) → (1r‘𝑅) ≠ (0g‘𝑅))) |
36 | 7, 35 | mpd 15 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → (1r‘𝑅) ≠ (0g‘𝑅)) |
37 | 15, 14 | isnzr 20745 | . . . 4 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
38 | 3, 36, 37 | sylanbrc 584 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ NzRing) → 𝑅 ∈ NzRing) |
39 | 38 | ex 414 | . 2 ⊢ (𝑅 ∈ Ring → (𝑃 ∈ NzRing → 𝑅 ∈ NzRing)) |
40 | 2, 39 | impbid2 225 | 1 ⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ 𝑃 ∈ NzRing)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 {crab 3406 Vcvv 3444 ifcif 4487 {csn 4587 ↦ cmpt 5189 × cxp 5632 ◡ccnv 5633 “ cima 5637 Oncon0 6318 ‘cfv 6497 (class class class)co 7358 1oc1o 8406 ↑m cmap 8768 Fincfn 8886 0cc0 11056 ℕcn 12158 ℕ0cn0 12418 0gc0g 17326 Grpcgrp 18753 1rcur 19918 Ringcrg 19969 NzRingcnzr 20743 mPoly cmpl 21324 Poly1cpl1 21564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-ofr 7619 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-hom 17162 df-cco 17163 df-0g 17328 df-gsum 17329 df-prds 17334 df-pws 17336 df-mre 17471 df-mrc 17472 df-acs 17474 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mulg 18878 df-subg 18930 df-ghm 19011 df-cntz 19102 df-cmn 19569 df-abl 19570 df-mgp 19902 df-ur 19919 df-ring 19971 df-subrg 20234 df-lmod 20338 df-lss 20408 df-nzr 20744 df-ascl 21277 df-psr 21327 df-mvr 21328 df-mpl 21329 df-opsr 21331 df-psr1 21567 df-vr1 21568 df-ply1 21569 df-coe1 21570 |
This theorem is referenced by: (None) |
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