Theorem bezoutlema 11932

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 9217	. . 3 |- 1 e. ZZ
2		0z 9202	. . 3 |- 0 e. ZZ
3		bezoutlema.b	. . . . . . 7 |- ( th -> B e. NN0 )
4	3	nn0cnd 9169	. . . . . 6 |- ( th -> B e. CC )
5	4	mul01d 8291	. . . . 5 |- ( th -> ( B x. 0 ) = 0 )
6	5	oveq2d 5858	. . . 4 |- ( th -> ( ( A x. 1 ) + ( B x. 0 ) ) = ( ( A x. 1 ) + 0 ) )
7		bezoutlema.a	. . . . . . 7 |- ( th -> A e. NN0 )
8	7	nn0cnd 9169	. . . . . 6 |- ( th -> A e. CC )
9		1cnd 7915	. . . . . 6 |- ( th -> 1 e. CC )
10	8, 9	mulcld 7919	. . . . 5 |- ( th -> ( A x. 1 ) e. CC )
11	10	addid1d 8047	. . . 4 |- ( th -> ( ( A x. 1 ) + 0 ) = ( A x. 1 ) )
12	8	mulid1d 7916	. . . 4 |- ( th -> ( A x. 1 ) = A )
13	6, 11, 12	3eqtrrd 2203	. . 3 |- ( th -> A = ( ( A x. 1 ) + ( B x. 0 ) ) )
14		oveq2 5850	. . . . . 6 |- ( s = 1 -> ( A x. s ) = ( A x. 1 ) )
15	14	oveq1d 5857	. . . . 5 |- ( s = 1 -> ( ( A x. s ) + ( B x. t ) ) = ( ( A x. 1 ) + ( B x. t ) ) )
16	15	eqeq2d 2177	. . . 4 |- ( s = 1 -> ( A = ( ( A x. s ) + ( B x. t ) ) <-> A = ( ( A x. 1 ) + ( B x. t ) ) ) )
17		oveq2 5850	. . . . . 6 |- ( t = 0 -> ( B x. t ) = ( B x. 0 ) )
18	17	oveq2d 5858	. . . . 5 |- ( t = 0 -> ( ( A x. 1 ) + ( B x. t ) ) = ( ( A x. 1 ) + ( B x. 0 ) ) )
19	18	eqeq2d 2177	. . . 4 |- ( t = 0 -> ( A = ( ( A x. 1 ) + ( B x. t ) ) <-> A = ( ( A x. 1 ) + ( B x. 0 ) ) ) )
20	16, 19	rspc2ev 2845	. . 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 1331	. 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 2172	. . . . . 6 |- ( r = A -> ( r = ( ( A x. s ) + ( B x. t ) ) <-> A = ( ( A x. s ) + ( B x. t ) ) ) )
24	23	2rexbidv 2491	. . . . 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 2983	. . 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 1343   e. wcel 2136   E.wrex 2445   [.wsbc 2951  (class class class)co 5842   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   NN0cn0 9114   ZZcz 9191

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192

This theorem is referenced by:  bezoutlemex  11934