Theorem bezoutlemb 11082

Description: Lemma for Bézout's identity. The is-bezout condition is satisfied by B. (Contributed by Jim Kingdon, 30-Dec-2021.)

Hypotheses

Ref	Expression
bezoutlema.is-bezout	|- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )
bezoutlema.a	|- ( th -> A e. NN0 )
bezoutlema.b	|- ( th -> B e. NN0 )

Assertion

Ref	Expression
bezoutlemb	|- ( th -> [. B / r ]. ph )

Distinct variable groups:   A, r, s, t   B, r, s, t

Allowed substitution hints:   ph( t, s, r)   th( t, s, r)

Proof of Theorem bezoutlemb

Step	Hyp	Ref	Expression
1		0z 8731	. . 3 |- 0 e. ZZ
2		1z 8746	. . 3 |- 1 e. ZZ
3		bezoutlema.a	. . . . . . 7 |- ( th -> A e. NN0 )
4	3	nn0cnd 8698	. . . . . 6 |- ( th -> A e. CC )
5	4	mul01d 7850	. . . . 5 |- ( th -> ( A x. 0 ) = 0 )
6	5	oveq1d 5649	. . . 4 |- ( th -> ( ( A x. 0 ) + ( B x. 1 ) ) = ( 0 + ( B x. 1 ) ) )
7		bezoutlema.b	. . . . . . 7 |- ( th -> B e. NN0 )
8	7	nn0cnd 8698	. . . . . 6 |- ( th -> B e. CC )
9		1cnd 7483	. . . . . 6 |- ( th -> 1 e. CC )
10	8, 9	mulcld 7487	. . . . 5 |- ( th -> ( B x. 1 ) e. CC )
11	10	addid2d 7611	. . . 4 |- ( th -> ( 0 + ( B x. 1 ) ) = ( B x. 1 ) )
12	8	mulid1d 7484	. . . 4 |- ( th -> ( B x. 1 ) = B )
13	6, 11, 12	3eqtrrd 2125	. . 3 |- ( th -> B = ( ( A x. 0 ) + ( B x. 1 ) ) )
14		oveq2 5642	. . . . . 6 |- ( s = 0 -> ( A x. s ) = ( A x. 0 ) )
15	14	oveq1d 5649	. . . . 5 |- ( s = 0 -> ( ( A x. s ) + ( B x. t ) ) = ( ( A x. 0 ) + ( B x. t ) ) )
16	15	eqeq2d 2099	. . . 4 |- ( s = 0 -> ( B = ( ( A x. s ) + ( B x. t ) ) <-> B = ( ( A x. 0 ) + ( B x. t ) ) ) )
17		oveq2 5642	. . . . . 6 |- ( t = 1 -> ( B x. t ) = ( B x. 1 ) )
18	17	oveq2d 5650	. . . . 5 |- ( t = 1 -> ( ( A x. 0 ) + ( B x. t ) ) = ( ( A x. 0 ) + ( B x. 1 ) ) )
19	18	eqeq2d 2099	. . . 4 |- ( t = 1 -> ( B = ( ( A x. 0 ) + ( B x. t ) ) <-> B = ( ( A x. 0 ) + ( B x. 1 ) ) ) )
20	16, 19	rspc2ev 2735	. . 3 |- ( ( 0 e. ZZ /\ 1 e. ZZ /\ B = ( ( A x. 0 ) + ( B x. 1 ) ) ) -> E. s e. ZZ E. t e. ZZ B = ( ( A x. s ) + ( B x. t ) ) )
21	1, 2, 13, 20	mp3an12i 1277	. 2 |- ( th -> E. s e. ZZ E. t e. ZZ B = ( ( A x. s ) + ( B x. t ) ) )
22		bezoutlema.is-bezout	. . . . 5 |- ( ph <-> E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) )
23		eqeq1 2094	. . . . . 6 |- ( r = B -> ( r = ( ( A x. s ) + ( B x. t ) ) <-> B = ( ( A x. s ) + ( B x. t ) ) ) )
24	23	2rexbidv 2403	. . . . 5 |- ( r = B -> ( E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) <-> E. s e. ZZ E. t e. ZZ B = ( ( A x. s ) + ( B x. t ) ) ) )
25	22, 24	syl5bb 190	. . . 4 |- ( r = B -> ( ph <-> E. s e. ZZ E. t e. ZZ B = ( ( A x. s ) + ( B x. t ) ) ) )
26	25	sbcieg 2869	. . 3 |- ( B e. NN0 -> ( [. B / r ]. ph <-> E. s e. ZZ E. t e. ZZ B = ( ( A x. s ) + ( B x. t ) ) ) )
27	7, 26	syl 14	. 2 |- ( th -> ( [. B / r ]. ph <-> E. s e. ZZ E. t e. ZZ B = ( ( A x. s ) + ( B x. t ) ) ) )
28	21, 27	mpbird 165	1 |- ( th -> [. B / r ]. ph )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   e. wcel 1438   E.wrex 2360   [.wsbc 2838  (class class class)co 5634   0cc0 7329   1c1 7330   + caddc 7332   x. cmul 7334   NN0cn0 8643   ZZcz 8720

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440

This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721

This theorem is referenced by:  bezoutlemex  11083