Theorem deg1ldg 23852

Description: A nonzero univariate polynomial always has a nonzero leading coefficient. (Contributed by Stefan O'Rear, 23-Mar-2015.)

Hypotheses

Ref	Expression
deg1z.d	|- D = ( deg1 ` R )
deg1z.p	|- P = (Poly1 ` R )
deg1z.z	|- .0. = ( 0g ` P )
deg1nn0cl.b	|- B = ( Base ` P )
deg1ldg.y	|- Y = ( 0g ` R )
deg1ldg.a	|- A = (coe1 ` F )

Assertion

Ref	Expression
deg1ldg	|- ( ( R e. Ring /\ F e. B /\ F =/= .0. ) -> ( A ` ( D ` F ) ) =/= Y )

Proof of Theorem deg1ldg

Dummy variables b d a c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		deg1z.d	. . . 4 |- D = ( deg1 ` R )
2	1	deg1fval 23840	. . 3 |- D = ( 1o mDeg R )
3		eqid 2622	. . 3 |- ( 1o mPoly R ) = ( 1o mPoly R )
4		deg1z.p	. . . 4 |- P = (Poly1 ` R )
5		eqid 2622	. . . 4 |- (PwSer1 ` R ) = (PwSer1 ` R )
6		deg1nn0cl.b	. . . 4 |- B = ( Base ` P )
7	4, 5, 6	ply1bas 19565	. . 3 |- B = ( Base ` ( 1o mPoly R ) )
8		deg1ldg.y	. . 3 |- Y = ( 0g ` R )
9		psr1baslem 19555	. . 3 |- ( NN0 ^m 1o ) = { c e. ( NN0 ^m 1o ) | ( `' c " NN ) e. Fin }
10		tdeglem2 23821	. . 3 |- ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) = ( a e. ( NN0 ^m 1o ) |-> (ℂfld gsumg a ) )
11		deg1z.z	. . . 4 |- .0. = ( 0g ` P )
12	3, 4, 11	ply1mpl0 19625	. . 3 |- .0. = ( 0g ` ( 1o mPoly R ) )
13	2, 3, 7, 8, 9, 10, 12	mdegldg 23826	. 2 |- ( ( R e. Ring /\ F e. B /\ F =/= .0. ) -> E. b e. ( NN0 ^m 1o ) ( ( F ` b ) =/= Y /\ ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) ) )
14		deg1ldg.a	. . . . . . . . . . 11 |- A = (coe1 ` F )
15	14	fvcoe1 19577	. . . . . . . . . 10 |- ( ( F e. B /\ b e. ( NN0 ^m 1o ) ) -> ( F ` b ) = ( A ` ( b ` (/) ) ) )
16	15	3ad2antl2 1224	. . . . . . . . 9 |- ( ( ( R e. Ring /\ F e. B /\ F =/= .0. ) /\ b e. ( NN0 ^m 1o ) ) -> ( F ` b ) = ( A ` ( b ` (/) ) ) )
17		fveq1 6190	. . . . . . . . . . . 12 |- ( a = b -> ( a ` (/) ) = ( b ` (/) ) )
18		eqid 2622	. . . . . . . . . . . 12 |- ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) = ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) )
19		fvex 6201	. . . . . . . . . . . 12 |- ( b ` (/) ) e. _V
20	17, 18, 19	fvmpt 6282	. . . . . . . . . . 11 |- ( b e. ( NN0 ^m 1o ) -> ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( b ` (/) ) )
21	20	fveq2d 6195	. . . . . . . . . 10 |- ( b e. ( NN0 ^m 1o ) -> ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) = ( A ` ( b ` (/) ) ) )
22	21	adantl 482	. . . . . . . . 9 |- ( ( ( R e. Ring /\ F e. B /\ F =/= .0. ) /\ b e. ( NN0 ^m 1o ) ) -> ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) = ( A ` ( b ` (/) ) ) )
23	16, 22	eqtr4d 2659	. . . . . . . 8 |- ( ( ( R e. Ring /\ F e. B /\ F =/= .0. ) /\ b e. ( NN0 ^m 1o ) ) -> ( F ` b ) = ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) )
24	23	neeq1d 2853	. . . . . . 7 |- ( ( ( R e. Ring /\ F e. B /\ F =/= .0. ) /\ b e. ( NN0 ^m 1o ) ) -> ( ( F ` b ) =/= Y <-> ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) =/= Y ) )
25	24	anbi1d 741	. . . . . 6 |- ( ( ( R e. Ring /\ F e. B /\ F =/= .0. ) /\ b e. ( NN0 ^m 1o ) ) -> ( ( ( F ` b ) =/= Y /\ ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) ) <-> ( ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) =/= Y /\ ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) ) ) )
26		ancom 466	. . . . . 6 |- ( ( ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) =/= Y /\ ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) ) <-> ( ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) /\ ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) =/= Y ) )
27	25, 26	syl6bb 276	. . . . 5 |- ( ( ( R e. Ring /\ F e. B /\ F =/= .0. ) /\ b e. ( NN0 ^m 1o ) ) -> ( ( ( F ` b ) =/= Y /\ ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) ) <-> ( ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) /\ ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) =/= Y ) ) )
28	27	rexbidva 3049	. . . 4 |- ( ( R e. Ring /\ F e. B /\ F =/= .0. ) -> ( E. b e. ( NN0 ^m 1o ) ( ( F ` b ) =/= Y /\ ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) ) <-> E. b e. ( NN0 ^m 1o ) ( ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) /\ ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) =/= Y ) ) )
29		df1o2 7572	. . . . . 6 |- 1o = { (/) }
30		nn0ex 11298	. . . . . 6 |- NN0 e. _V
31		0ex 4790	. . . . . 6 |- (/) e. _V
32	29, 30, 31, 18	mapsnf1o2 7905	. . . . 5 |- ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) : ( NN0 ^m 1o ) -1-1-onto-> NN0
33		f1ofo 6144	. . . . 5 |- ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) : ( NN0 ^m 1o ) -1-1-onto-> NN0 -> ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) : ( NN0 ^m 1o ) -onto-> NN0 )
34		eqeq1 2626	. . . . . . 7 |- ( ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = d -> ( ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) <-> d = ( D ` F ) ) )
35		fveq2 6191	. . . . . . . 8 |- ( ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = d -> ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) = ( A ` d ) )
36	35	neeq1d 2853	. . . . . . 7 |- ( ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = d -> ( ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) =/= Y <-> ( A ` d ) =/= Y ) )
37	34, 36	anbi12d 747	. . . . . 6 |- ( ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = d -> ( ( ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) /\ ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) =/= Y ) <-> ( d = ( D ` F ) /\ ( A ` d ) =/= Y ) ) )
38	37	cbvexfo 6545	. . . . 5 |- ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) : ( NN0 ^m 1o ) -onto-> NN0 -> ( E. b e. ( NN0 ^m 1o ) ( ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) /\ ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) =/= Y ) <-> E. d e. NN0 ( d = ( D ` F ) /\ ( A ` d ) =/= Y ) ) )
39	32, 33, 38	mp2b 10	. . . 4 |- ( E. b e. ( NN0 ^m 1o ) ( ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) /\ ( A ` ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) ) =/= Y ) <-> E. d e. NN0 ( d = ( D ` F ) /\ ( A ` d ) =/= Y ) )
40	28, 39	syl6bb 276	. . 3 |- ( ( R e. Ring /\ F e. B /\ F =/= .0. ) -> ( E. b e. ( NN0 ^m 1o ) ( ( F ` b ) =/= Y /\ ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) ) <-> E. d e. NN0 ( d = ( D ` F ) /\ ( A ` d ) =/= Y ) ) )
41	1, 4, 11, 6	deg1nn0cl 23848	. . . 4 |- ( ( R e. Ring /\ F e. B /\ F =/= .0. ) -> ( D ` F ) e. NN0 )
42		fveq2 6191	. . . . . 6 |- ( d = ( D ` F ) -> ( A ` d ) = ( A ` ( D ` F ) ) )
43	42	neeq1d 2853	. . . . 5 |- ( d = ( D ` F ) -> ( ( A ` d ) =/= Y <-> ( A ` ( D ` F ) ) =/= Y ) )
44	43	ceqsrexv 3336	. . . 4 |- ( ( D ` F ) e. NN0 -> ( E. d e. NN0 ( d = ( D ` F ) /\ ( A ` d ) =/= Y ) <-> ( A ` ( D ` F ) ) =/= Y ) )
45	41, 44	syl 17	. . 3 |- ( ( R e. Ring /\ F e. B /\ F =/= .0. ) -> ( E. d e. NN0 ( d = ( D ` F ) /\ ( A ` d ) =/= Y ) <-> ( A ` ( D ` F ) ) =/= Y ) )
46	40, 45	bitrd 268	. 2 |- ( ( R e. Ring /\ F e. B /\ F =/= .0. ) -> ( E. b e. ( NN0 ^m 1o ) ( ( F ` b ) =/= Y /\ ( ( a e. ( NN0 ^m 1o ) |-> ( a ` (/) ) ) ` b ) = ( D ` F ) ) <-> ( A ` ( D ` F ) ) =/= Y ) )
47	13, 46	mpbid 222	1 |- ( ( R e. Ring /\ F e. B /\ F =/= .0. ) -> ( A ` ( D ` F ) ) =/= Y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   (/)c0 3915   |-> cmpt 4729   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1oc1o 7553   ^m cmap 7857   NN0cn0 11292   Basecbs 15857   0gc0g 16100   Ringcrg 18547   mPoly cmpl 19353  PwSer₁cps1 19545  Poly₁cpl1 19547  coe₁cco1 19548   deg₁ cdg1 23814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-psr 19356  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-ply1 19552  df-coe1 19553  df-cnfld 19747  df-mdeg 23815  df-deg1 23816

This theorem is referenced by:  deg1ldgn  23853  deg1ldgdomn  23854  deg1add  23863  deg1mul2  23874  drnguc1p  23930