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Mirrors > Home > MPE Home > Th. List > ply1coe1eq | Structured version Visualization version GIF version |
Description: Two polynomials over the same ring are equal iff they have identical coefficients. (Contributed by AV, 13-Oct-2019.) |
Ref | Expression |
---|---|
eqcoe1ply1eq.p | ⊢ 𝑃 = (Poly1‘𝑅) |
eqcoe1ply1eq.b | ⊢ 𝐵 = (Base‘𝑃) |
eqcoe1ply1eq.a | ⊢ 𝐴 = (coe1‘𝐾) |
eqcoe1ply1eq.c | ⊢ 𝐶 = (coe1‘𝐿) |
Ref | Expression |
---|---|
ply1coe1eq | ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∀𝑘 ∈ ℕ0 (𝐴‘𝑘) = (𝐶‘𝑘) ↔ 𝐾 = 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcoe1ply1eq.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | eqcoe1ply1eq.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
3 | eqcoe1ply1eq.a | . . 3 ⊢ 𝐴 = (coe1‘𝐾) | |
4 | eqcoe1ply1eq.c | . . 3 ⊢ 𝐶 = (coe1‘𝐿) | |
5 | 1, 2, 3, 4 | eqcoe1ply1eq 21684 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∀𝑘 ∈ ℕ0 (𝐴‘𝑘) = (𝐶‘𝑘) → 𝐾 = 𝐿)) |
6 | fveq2 6843 | . . . . . . . 8 ⊢ (𝐾 = 𝐿 → (coe1‘𝐾) = (coe1‘𝐿)) | |
7 | 6 | adantl 483 | . . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝐾 = 𝐿) → (coe1‘𝐾) = (coe1‘𝐿)) |
8 | 7, 3, 4 | 3eqtr4g 2798 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝐾 = 𝐿) → 𝐴 = 𝐶) |
9 | 8 | adantr 482 | . . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝐾 = 𝐿) ∧ 𝑘 ∈ ℕ0) → 𝐴 = 𝐶) |
10 | 9 | fveq1d 6845 | . . . 4 ⊢ ((((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝐾 = 𝐿) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) = (𝐶‘𝑘)) |
11 | 10 | ralrimiva 3140 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) ∧ 𝐾 = 𝐿) → ∀𝑘 ∈ ℕ0 (𝐴‘𝑘) = (𝐶‘𝑘)) |
12 | 11 | ex 414 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (𝐾 = 𝐿 → ∀𝑘 ∈ ℕ0 (𝐴‘𝑘) = (𝐶‘𝑘))) |
13 | 5, 12 | impbid 211 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵) → (∀𝑘 ∈ ℕ0 (𝐴‘𝑘) = (𝐶‘𝑘) ↔ 𝐾 = 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ‘cfv 6497 ℕ0cn0 12418 Basecbs 17088 Ringcrg 19969 Poly1cpl1 21564 coe1cco1 21565 |
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-srg 19923 df-ring 19971 df-subrg 20234 df-lmod 20338 df-lss 20408 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: gsumply1eq 21692 m2cpminvid2 22120 pm2mpf1 22164 pm2mpmhmlem2 22184 cayleyhamilton1 22257 ply1scleq 32315 ply1moneq 32335 ply1gsumz 32339 ply1mulgsum 46557 |
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