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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 6897 | . . . . 5 β’ (π = 0 β ((coe1β(π΄βπΈ))βπ) = ((coe1β(π΄βπΈ))β0)) | |
2 | fveq2 6897 | . . . . 5 β’ (π = 0 β ((coe1β(π΄βπΉ))βπ) = ((coe1β(π΄βπΉ))β0)) | |
3 | 1, 2 | eqeq12d 2744 | . . . 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 2728 | . . . . . . . 8 β’ (Baseβπ) = (Baseβπ) | |
10 | 6, 7, 8, 9 | ply1sclcl 22205 | . . . . . . 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 22205 | . . . . . . 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 2728 | . . . . . . 7 β’ (coe1β(π΄βπΈ)) = (coe1β(π΄βπΈ)) | |
16 | eqid 2728 | . . . . . . 7 β’ (coe1β(π΄βπΉ)) = (coe1β(π΄βπΉ)) | |
17 | 6, 9, 15, 16 | ply1coe1eq 22219 | . . . . . 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 1369 | . . . . 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 12518 | . . . . 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 3610 | . . 3 β’ ((π β§ (π΄βπΈ) = (π΄βπΉ)) β ((coe1β(π΄βπΈ))β0) = ((coe1β(π΄βπΉ))β0)) |
23 | 6, 7, 8 | ply1sclid 22207 | . . . . 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 22207 | . . . . 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 2778 | . 2 β’ ((π β§ (π΄βπΈ) = (π΄βπΉ)) β πΈ = πΉ) |
30 | fveq2 6897 | . . 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 800 | 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 1534 β wcel 2099 βwral 3058 βcfv 6548 0cc0 11139 β0cn0 12503 Basecbs 17180 Ringcrg 20173 algSccascl 21786 Poly1cpl1 22096 coe1cco1 22097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-srg 20127 df-ring 20175 df-subrng 20483 df-subrg 20508 df-lmod 20745 df-lss 20816 df-ascl 21789 df-psr 21842 df-mvr 21843 df-mpl 21844 df-opsr 21846 df-psr1 22099 df-vr1 22100 df-ply1 22101 df-coe1 22102 |
This theorem is referenced by: ply1chr 22225 |
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