Theorem bezoutlemb 12028

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 9289	. . 3 |- 0 e. ZZ
2		1z 9304	. . 3 |- 1 e. ZZ
3		bezoutlema.a	. . . . . . 7 |- ( th -> A e. NN0 )
4	3	nn0cnd 9256	. . . . . 6 |- ( th -> A e. CC )
5	4	mul01d 8375	. . . . 5 |- ( th -> ( A x. 0 ) = 0 )
6	5	oveq1d 5907	. . . 4 |- ( th -> ( ( A x. 0 ) + ( B x. 1 ) ) = ( 0 + ( B x. 1 ) ) )
7		bezoutlema.b	. . . . . . 7 |- ( th -> B e. NN0 )
8	7	nn0cnd 9256	. . . . . 6 |- ( th -> B e. CC )
9		1cnd 7998	. . . . . 6 |- ( th -> 1 e. CC )
10	8, 9	mulcld 8003	. . . . 5 |- ( th -> ( B x. 1 ) e. CC )
11	10	addlidd 8132	. . . 4 |- ( th -> ( 0 + ( B x. 1 ) ) = ( B x. 1 ) )
12	8	mulridd 7999	. . . 4 |- ( th -> ( B x. 1 ) = B )
13	6, 11, 12	3eqtrrd 2227	. . 3 |- ( th -> B = ( ( A x. 0 ) + ( B x. 1 ) ) )
14		oveq2 5900	. . . . . 6 |- ( s = 0 -> ( A x. s ) = ( A x. 0 ) )
15	14	oveq1d 5907	. . . . 5 |- ( s = 0 -> ( ( A x. s ) + ( B x. t ) ) = ( ( A x. 0 ) + ( B x. t ) ) )
16	15	eqeq2d 2201	. . . 4 |- ( s = 0 -> ( B = ( ( A x. s ) + ( B x. t ) ) <-> B = ( ( A x. 0 ) + ( B x. t ) ) ) )
17		oveq2 5900	. . . . . 6 |- ( t = 1 -> ( B x. t ) = ( B x. 1 ) )
18	17	oveq2d 5908	. . . . 5 |- ( t = 1 -> ( ( A x. 0 ) + ( B x. t ) ) = ( ( A x. 0 ) + ( B x. 1 ) ) )
19	18	eqeq2d 2201	. . . 4 |- ( t = 1 -> ( B = ( ( A x. 0 ) + ( B x. t ) ) <-> B = ( ( A x. 0 ) + ( B x. 1 ) ) ) )
20	16, 19	rspc2ev 2871	. . 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 1352	. 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 2196	. . . . . 6 |- ( r = B -> ( r = ( ( A x. s ) + ( B x. t ) ) <-> B = ( ( A x. s ) + ( B x. t ) ) ) )
24	23	2rexbidv 2515	. . . . 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	bitrid 192	. . . 4 |- ( r = B -> ( ph <-> E. s e. ZZ E. t e. ZZ B = ( ( A x. s ) + ( B x. t ) ) ) )
26	25	sbcieg 3010	. . 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 167	1 |- ( th -> [. B / r ]. ph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2160   E.wrex 2469   [.wsbc 2977  (class class class)co 5892   0cc0 7836   1c1 7837   + caddc 7839   x. cmul 7841   NN0cn0 9201   ZZcz 9278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-ltadd 7952

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5234  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-inn 8945  df-n0 9202  df-z 9279

This theorem is referenced by:  bezoutlemex  12029