Theorem plydivalg 24054

Description: The division algorithm on polynomials over a subfield S of the complex numbers. If F and G =/= 0 are polynomials over S, then there is a unique quotient polynomial q such that the remainder F - G x. q is either zero or has degree less than G. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypotheses

Ref	Expression
plydiv.pl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
plydiv.tm	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
plydiv.rc	|- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )
plydiv.m1	|- ( ph -> -u 1 e. S )
plydiv.f	|- ( ph -> F e. (Poly ` S ) )
plydiv.g	|- ( ph -> G e. (Poly ` S ) )
plydiv.z	|- ( ph -> G =/= 0p )
plydiv.r	|- R = ( F oF - ( G oF x. q ) )

Assertion

Ref	Expression
plydivalg	|- ( ph -> E! q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )

Distinct variable groups:   x, y, q, F   ph, x, y   G, q, x, y   x, R, y   S, q, x, y   ph, q

Allowed substitution hint:   R( q)

Proof of Theorem plydivalg

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		plydiv.pl	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
2		plydiv.tm	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
3		plydiv.rc	. . 3 |- ( ( ph /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )
4		plydiv.m1	. . 3 |- ( ph -> -u 1 e. S )
5		plydiv.f	. . 3 |- ( ph -> F e. (Poly ` S ) )
6		plydiv.g	. . 3 |- ( ph -> G e. (Poly ` S ) )
7		plydiv.z	. . 3 |- ( ph -> G =/= 0p )
8		plydiv.r	. . 3 |- R = ( F oF - ( G oF x. q ) )
9	1, 2, 3, 4, 5, 6, 7, 8	plydivex 24052	. 2 |- ( ph -> E. q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )
10		simpll 790	. . . . . 6 |- ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) -> ph )
11	10, 1	sylan 488	. . . . 5 |- ( ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )
12	10, 2	sylan 488	. . . . 5 |- ( ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )
13	10, 3	sylan 488	. . . . 5 |- ( ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) /\ ( x e. S /\ x =/= 0 ) ) -> ( 1 / x ) e. S )
14	10, 4	syl 17	. . . . 5 |- ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) -> -u 1 e. S )
15	10, 5	syl 17	. . . . 5 |- ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) -> F e. (Poly ` S ) )
16	10, 6	syl 17	. . . . 5 |- ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) -> G e. (Poly ` S ) )
17	10, 7	syl 17	. . . . 5 |- ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) -> G =/= 0p )
18		eqid 2622	. . . . 5 |- ( F oF - ( G oF x. p ) ) = ( F oF - ( G oF x. p ) )
19		simplrr 801	. . . . 5 |- ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) -> p e. (Poly ` S ) )
20		simprr 796	. . . . 5 |- ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) -> ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) )
21		simplrl 800	. . . . 5 |- ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) -> q e. (Poly ` S ) )
22		simprl 794	. . . . 5 |- ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) -> ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )
23	11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 8, 21, 22	plydiveu 24053	. . . 4 |- ( ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) /\ ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) ) -> q = p )
24	23	ex 450	. . 3 |- ( ( ph /\ ( q e. (Poly ` S ) /\ p e. (Poly ` S ) ) ) -> ( ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) -> q = p ) )
25	24	ralrimivva 2971	. 2 |- ( ph -> A. q e. (Poly ` S ) A. p e. (Poly ` S ) ( ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) -> q = p ) )
26		oveq2 6658	. . . . . . 7 |- ( q = p -> ( G oF x. q ) = ( G oF x. p ) )
27	26	oveq2d 6666	. . . . . 6 |- ( q = p -> ( F oF - ( G oF x. q ) ) = ( F oF - ( G oF x. p ) ) )
28	8, 27	syl5eq 2668	. . . . 5 |- ( q = p -> R = ( F oF - ( G oF x. p ) ) )
29	28	eqeq1d 2624	. . . 4 |- ( q = p -> ( R = 0p <-> ( F oF - ( G oF x. p ) ) = 0p ) )
30	28	fveq2d 6195	. . . . 5 |- ( q = p -> (deg ` R ) = (deg ` ( F oF - ( G oF x. p ) ) ) )
31	30	breq1d 4663	. . . 4 |- ( q = p -> ( (deg ` R ) < (deg ` G ) <-> (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) )
32	29, 31	orbi12d 746	. . 3 |- ( q = p -> ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) <-> ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) )
33	32	reu4 3400	. 2 |- ( E! q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) <-> ( E. q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ A. q e. (Poly ` S ) A. p e. (Poly ` S ) ( ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) /\ ( ( F oF - ( G oF x. p ) ) = 0p \/ (deg ` ( F oF - ( G oF x. p ) ) ) < (deg ` G ) ) ) -> q = p ) ) )
34	9, 25, 33	sylanbrc 698	1 |- ( ph -> E! q e. (Poly ` S ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   oFcof 6895   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   0pc0p 23436  Polycply 23940  degcdgr 23943

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-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  quotlem  24055