Theorem quotval 24047

Description: Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)

Hypothesis

Ref	Expression
quotval.1	|- R = ( F oF - ( G oF x. q ) )

Assertion

Ref	Expression
quotval	|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( F quot G ) = ( iota_ q e. (Poly ` CC ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )

Distinct variable groups:   F, q   G, q

Allowed substitution hints:   R( q)   S( q)

Proof of Theorem quotval

Dummy variables f g r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyssc 23956	. . 3 |- (Poly ` S ) C_ (Poly ` CC )
2	1	sseli 3599	. 2 |- ( F e. (Poly ` S ) -> F e. (Poly ` CC ) )
3	1	sseli 3599	. . 3 |- ( G e. (Poly ` S ) -> G e. (Poly ` CC ) )
4		eldifsn 4317	. . . . 5 |- ( G e. ( (Poly ` CC ) \ { 0p } ) <-> ( G e. (Poly ` CC ) /\ G =/= 0p ) )
5		oveq1 6657	. . . . . . . . . . 11 |- ( g = G -> ( g oF x. q ) = ( G oF x. q ) )
6		oveq12 6659	. . . . . . . . . . 11 |- ( ( f = F /\ ( g oF x. q ) = ( G oF x. q ) ) -> ( f oF - ( g oF x. q ) ) = ( F oF - ( G oF x. q ) ) )
7	5, 6	sylan2 491	. . . . . . . . . 10 |- ( ( f = F /\ g = G ) -> ( f oF - ( g oF x. q ) ) = ( F oF - ( G oF x. q ) ) )
8		quotval.1	. . . . . . . . . 10 |- R = ( F oF - ( G oF x. q ) )
9	7, 8	syl6eqr 2674	. . . . . . . . 9 |- ( ( f = F /\ g = G ) -> ( f oF - ( g oF x. q ) ) = R )
10	9	sbceq1d 3440	. . . . . . . 8 |- ( ( f = F /\ g = G ) -> ( [. ( f oF - ( g oF x. q ) ) / r ]. ( r = 0p \/ (deg ` r ) < (deg ` g ) ) <-> [. R / r ]. ( r = 0p \/ (deg ` r ) < (deg ` g ) ) ) )
11		ovex 6678	. . . . . . . . . . 11 |- ( F oF - ( G oF x. q ) ) e. _V
12	8, 11	eqeltri 2697	. . . . . . . . . 10 |- R e. _V
13		eqeq1 2626	. . . . . . . . . . 11 |- ( r = R -> ( r = 0p <-> R = 0p ) )
14		fveq2 6191	. . . . . . . . . . . 12 |- ( r = R -> (deg ` r ) = (deg ` R ) )
15	14	breq1d 4663	. . . . . . . . . . 11 |- ( r = R -> ( (deg ` r ) < (deg ` g ) <-> (deg ` R ) < (deg ` g ) ) )
16	13, 15	orbi12d 746	. . . . . . . . . 10 |- ( r = R -> ( ( r = 0p \/ (deg ` r ) < (deg ` g ) ) <-> ( R = 0p \/ (deg ` R ) < (deg ` g ) ) ) )
17	12, 16	sbcie 3470	. . . . . . . . 9 |- ( [. R / r ]. ( r = 0p \/ (deg ` r ) < (deg ` g ) ) <-> ( R = 0p \/ (deg ` R ) < (deg ` g ) ) )
18		simpr 477	. . . . . . . . . . . 12 |- ( ( f = F /\ g = G ) -> g = G )
19	18	fveq2d 6195	. . . . . . . . . . 11 |- ( ( f = F /\ g = G ) -> (deg ` g ) = (deg ` G ) )
20	19	breq2d 4665	. . . . . . . . . 10 |- ( ( f = F /\ g = G ) -> ( (deg ` R ) < (deg ` g ) <-> (deg ` R ) < (deg ` G ) ) )
21	20	orbi2d 738	. . . . . . . . 9 |- ( ( f = F /\ g = G ) -> ( ( R = 0p \/ (deg ` R ) < (deg ` g ) ) <-> ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )
22	17, 21	syl5bb 272	. . . . . . . 8 |- ( ( f = F /\ g = G ) -> ( [. R / r ]. ( r = 0p \/ (deg ` r ) < (deg ` g ) ) <-> ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )
23	10, 22	bitrd 268	. . . . . . 7 |- ( ( f = F /\ g = G ) -> ( [. ( f oF - ( g oF x. q ) ) / r ]. ( r = 0p \/ (deg ` r ) < (deg ` g ) ) <-> ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )
24	23	riotabidv 6613	. . . . . 6 |- ( ( f = F /\ g = G ) -> ( iota_ q e. (Poly ` CC ) [. ( f oF - ( g oF x. q ) ) / r ]. ( r = 0p \/ (deg ` r ) < (deg ` g ) ) ) = ( iota_ q e. (Poly ` CC ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )
25		df-quot 24046	. . . . . 6 |- quot = ( f e. (Poly ` CC ) , g e. ( (Poly ` CC ) \ { 0p } ) |-> ( iota_ q e. (Poly ` CC ) [. ( f oF - ( g oF x. q ) ) / r ]. ( r = 0p \/ (deg ` r ) < (deg ` g ) ) ) )
26		riotaex 6615	. . . . . 6 |- ( iota_ q e. (Poly ` CC ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) e. _V
27	24, 25, 26	ovmpt2a 6791	. . . . 5 |- ( ( F e. (Poly ` CC ) /\ G e. ( (Poly ` CC ) \ { 0p } ) ) -> ( F quot G ) = ( iota_ q e. (Poly ` CC ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )
28	4, 27	sylan2br 493	. . . 4 |- ( ( F e. (Poly ` CC ) /\ ( G e. (Poly ` CC ) /\ G =/= 0p ) ) -> ( F quot G ) = ( iota_ q e. (Poly ` CC ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )
29	28	3impb 1260	. . 3 |- ( ( F e. (Poly ` CC ) /\ G e. (Poly ` CC ) /\ G =/= 0p ) -> ( F quot G ) = ( iota_ q e. (Poly ` CC ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )
30	3, 29	syl3an2 1360	. 2 |- ( ( F e. (Poly ` CC ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( F quot G ) = ( iota_ q e. (Poly ` CC ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )
31	2, 30	syl3an1 1359	1 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( F quot G ) = ( iota_ q e. (Poly ` CC ) ( R = 0p \/ (deg ` R ) < (deg ` G ) ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   {csn 4177   class class class wbr 4653   ` cfv 5888   iota_crio 6610  (class class class)co 6650   oFcof 6895   CCcc 9934   x. cmul 9941   < clt 10074   - cmin 10266   0pc0p 23436  Polycply 23940  degcdgr 23943   quot cquot 24045

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-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

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-ral 2917  df-rex 2918  df-reu 2919  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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-map 7859  df-nn 11021  df-n0 11293  df-ply 23944  df-quot 24046

This theorem is referenced by:  quotlem  24055