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Mirrors > Home > MPE Home > Th. List > ply1scleq | Structured version Visualization version GIF version |
Description: Equality of a constant polynomial is the same as equality of the constant term. (Contributed by Thierry Arnoux, 24-Jul-2024.) |
Ref | Expression |
---|---|
ply1scleq.p | β’ π = (Poly1βπ ) |
ply1scleq.b | β’ π΅ = (Baseβπ ) |
ply1scleq.a | β’ π΄ = (algScβπ) |
ply1scleq.r | β’ (π β π β Ring) |
ply1scleq.e | β’ (π β πΈ β π΅) |
ply1scleq.f | β’ (π β πΉ β π΅) |
Ref | Expression |
---|---|
ply1scleq | β’ (π β ((π΄βπΈ) = (π΄βπΉ) β πΈ = πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6882 | . . . . 5 β’ (π = 0 β ((coe1β(π΄βπΈ))βπ) = ((coe1β(π΄βπΈ))β0)) | |
2 | fveq2 6882 | . . . . 5 β’ (π = 0 β ((coe1β(π΄βπΉ))βπ) = ((coe1β(π΄βπΉ))β0)) | |
3 | 1, 2 | eqeq12d 2740 | . . . 4 β’ (π = 0 β (((coe1β(π΄βπΈ))βπ) = ((coe1β(π΄βπΉ))βπ) β ((coe1β(π΄βπΈ))β0) = ((coe1β(π΄βπΉ))β0))) |
4 | ply1scleq.r | . . . . . 6 β’ (π β π β Ring) | |
5 | ply1scleq.e | . . . . . . 7 β’ (π β πΈ β π΅) | |
6 | ply1scleq.p | . . . . . . . 8 β’ π = (Poly1βπ ) | |
7 | ply1scleq.a | . . . . . . . 8 β’ π΄ = (algScβπ) | |
8 | ply1scleq.b | . . . . . . . 8 β’ π΅ = (Baseβπ ) | |
9 | eqid 2724 | . . . . . . . 8 β’ (Baseβπ) = (Baseβπ) | |
10 | 6, 7, 8, 9 | ply1sclcl 22149 | . . . . . . 7 β’ ((π β Ring β§ πΈ β π΅) β (π΄βπΈ) β (Baseβπ)) |
11 | 4, 5, 10 | syl2anc 583 | . . . . . 6 β’ (π β (π΄βπΈ) β (Baseβπ)) |
12 | ply1scleq.f | . . . . . . 7 β’ (π β πΉ β π΅) | |
13 | 6, 7, 8, 9 | ply1sclcl 22149 | . . . . . . 7 β’ ((π β Ring β§ πΉ β π΅) β (π΄βπΉ) β (Baseβπ)) |
14 | 4, 12, 13 | syl2anc 583 | . . . . . 6 β’ (π β (π΄βπΉ) β (Baseβπ)) |
15 | eqid 2724 | . . . . . . 7 β’ (coe1β(π΄βπΈ)) = (coe1β(π΄βπΈ)) | |
16 | eqid 2724 | . . . . . . 7 β’ (coe1β(π΄βπΉ)) = (coe1β(π΄βπΉ)) | |
17 | 6, 9, 15, 16 | ply1coe1eq 22163 | . . . . . 6 β’ ((π β Ring β§ (π΄βπΈ) β (Baseβπ) β§ (π΄βπΉ) β (Baseβπ)) β (βπ β β0 ((coe1β(π΄βπΈ))βπ) = ((coe1β(π΄βπΉ))βπ) β (π΄βπΈ) = (π΄βπΉ))) |
18 | 4, 11, 14, 17 | syl3anc 1368 | . . . . 5 β’ (π β (βπ β β0 ((coe1β(π΄βπΈ))βπ) = ((coe1β(π΄βπΉ))βπ) β (π΄βπΈ) = (π΄βπΉ))) |
19 | 18 | biimpar 477 | . . . 4 β’ ((π β§ (π΄βπΈ) = (π΄βπΉ)) β βπ β β0 ((coe1β(π΄βπΈ))βπ) = ((coe1β(π΄βπΉ))βπ)) |
20 | 0nn0 12486 | . . . . 5 β’ 0 β β0 | |
21 | 20 | a1i 11 | . . . 4 β’ ((π β§ (π΄βπΈ) = (π΄βπΉ)) β 0 β β0) |
22 | 3, 19, 21 | rspcdva 3605 | . . 3 β’ ((π β§ (π΄βπΈ) = (π΄βπΉ)) β ((coe1β(π΄βπΈ))β0) = ((coe1β(π΄βπΉ))β0)) |
23 | 6, 7, 8 | ply1sclid 22151 | . . . . 5 β’ ((π β Ring β§ πΈ β π΅) β πΈ = ((coe1β(π΄βπΈ))β0)) |
24 | 4, 5, 23 | syl2anc 583 | . . . 4 β’ (π β πΈ = ((coe1β(π΄βπΈ))β0)) |
25 | 24 | adantr 480 | . . 3 β’ ((π β§ (π΄βπΈ) = (π΄βπΉ)) β πΈ = ((coe1β(π΄βπΈ))β0)) |
26 | 6, 7, 8 | ply1sclid 22151 | . . . . 5 β’ ((π β Ring β§ πΉ β π΅) β πΉ = ((coe1β(π΄βπΉ))β0)) |
27 | 4, 12, 26 | syl2anc 583 | . . . 4 β’ (π β πΉ = ((coe1β(π΄βπΉ))β0)) |
28 | 27 | adantr 480 | . . 3 β’ ((π β§ (π΄βπΈ) = (π΄βπΉ)) β πΉ = ((coe1β(π΄βπΉ))β0)) |
29 | 22, 25, 28 | 3eqtr4d 2774 | . 2 β’ ((π β§ (π΄βπΈ) = (π΄βπΉ)) β πΈ = πΉ) |
30 | fveq2 6882 | . . 3 β’ (πΈ = πΉ β (π΄βπΈ) = (π΄βπΉ)) | |
31 | 30 | adantl 481 | . 2 β’ ((π β§ πΈ = πΉ) β (π΄βπΈ) = (π΄βπΉ)) |
32 | 29, 31 | impbida 798 | 1 β’ (π β ((π΄βπΈ) = (π΄βπΉ) β πΈ = πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3053 βcfv 6534 0cc0 11107 β0cn0 12471 Basecbs 17149 Ringcrg 20134 algSccascl 21736 Poly1cpl1 22040 coe1cco1 22041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-ofr 7665 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-srg 20088 df-ring 20136 df-subrng 20442 df-subrg 20467 df-lmod 20704 df-lss 20775 df-ascl 21739 df-psr 21792 df-mvr 21793 df-mpl 21794 df-opsr 21796 df-psr1 22043 df-vr1 22044 df-ply1 22045 df-coe1 22046 |
This theorem is referenced by: ply1chr 22169 |
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