Theorem bezoutlema 11687

Description: Lemma for Bézout's identity. The is-bezout condition is satisfied by A. (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
bezoutlema	|- ( th -> [. A / 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 bezoutlema

Step	Hyp	Ref	Expression
1		1z 9080	. . 3 |- 1 e. ZZ
2		0z 9065	. . 3 |- 0 e. ZZ
3		bezoutlema.b	. . . . . . 7 |- ( th -> B e. NN0 )
4	3	nn0cnd 9032	. . . . . 6 |- ( th -> B e. CC )
5	4	mul01d 8155	. . . . 5 |- ( th -> ( B x. 0 ) = 0 )
6	5	oveq2d 5790	. . . 4 |- ( th -> ( ( A x. 1 ) + ( B x. 0 ) ) = ( ( A x. 1 ) + 0 ) )
7		bezoutlema.a	. . . . . . 7 |- ( th -> A e. NN0 )
8	7	nn0cnd 9032	. . . . . 6 |- ( th -> A e. CC )
9		1cnd 7782	. . . . . 6 |- ( th -> 1 e. CC )
10	8, 9	mulcld 7786	. . . . 5 |- ( th -> ( A x. 1 ) e. CC )
11	10	addid1d 7911	. . . 4 |- ( th -> ( ( A x. 1 ) + 0 ) = ( A x. 1 ) )
12	8	mulid1d 7783	. . . 4 |- ( th -> ( A x. 1 ) = A )
13	6, 11, 12	3eqtrrd 2177	. . 3 |- ( th -> A = ( ( A x. 1 ) + ( B x. 0 ) ) )
14		oveq2 5782	. . . . . 6 |- ( s = 1 -> ( A x. s ) = ( A x. 1 ) )
15	14	oveq1d 5789	. . . . 5 |- ( s = 1 -> ( ( A x. s ) + ( B x. t ) ) = ( ( A x. 1 ) + ( B x. t ) ) )
16	15	eqeq2d 2151	. . . 4 |- ( s = 1 -> ( A = ( ( A x. s ) + ( B x. t ) ) <-> A = ( ( A x. 1 ) + ( B x. t ) ) ) )
17		oveq2 5782	. . . . . 6 |- ( t = 0 -> ( B x. t ) = ( B x. 0 ) )
18	17	oveq2d 5790	. . . . 5 |- ( t = 0 -> ( ( A x. 1 ) + ( B x. t ) ) = ( ( A x. 1 ) + ( B x. 0 ) ) )
19	18	eqeq2d 2151	. . . 4 |- ( t = 0 -> ( A = ( ( A x. 1 ) + ( B x. t ) ) <-> A = ( ( A x. 1 ) + ( B x. 0 ) ) ) )
20	16, 19	rspc2ev 2804	. . 3 |- ( ( 1 e. ZZ /\ 0 e. ZZ /\ A = ( ( A x. 1 ) + ( B x. 0 ) ) ) -> E. s e. ZZ E. t e. ZZ A = ( ( A x. s ) + ( B x. t ) ) )
21	1, 2, 13, 20	mp3an12i 1319	. 2 |- ( th -> E. s e. ZZ E. t e. ZZ A = ( ( 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 2146	. . . . . 6 |- ( r = A -> ( r = ( ( A x. s ) + ( B x. t ) ) <-> A = ( ( A x. s ) + ( B x. t ) ) ) )
24	23	2rexbidv 2460	. . . . 5 |- ( r = A -> ( E. s e. ZZ E. t e. ZZ r = ( ( A x. s ) + ( B x. t ) ) <-> E. s e. ZZ E. t e. ZZ A = ( ( A x. s ) + ( B x. t ) ) ) )
25	22, 24	syl5bb 191	. . . 4 |- ( r = A -> ( ph <-> E. s e. ZZ E. t e. ZZ A = ( ( A x. s ) + ( B x. t ) ) ) )
26	25	sbcieg 2941	. . 3 |- ( A e. NN0 -> ( [. A / r ]. ph <-> E. s e. ZZ E. t e. ZZ A = ( ( A x. s ) + ( B x. t ) ) ) )
27	7, 26	syl 14	. 2 |- ( th -> ( [. A / r ]. ph <-> E. s e. ZZ E. t e. ZZ A = ( ( A x. s ) + ( B x. t ) ) ) )
28	21, 27	mpbird 166	1 |- ( th -> [. A / r ]. ph )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2417   [.wsbc 2909  (class class class)co 5774   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   NN0cn0 8977   ZZcz 9054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055

This theorem is referenced by:  bezoutlemex  11689