Theorem bezoutlemb 12137

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 9328	. . 3 |- 0 e. ZZ
2		1z 9343	. . 3 |- 1 e. ZZ
3		bezoutlema.a	. . . . . . 7 |- ( th -> A e. NN0 )
4	3	nn0cnd 9295	. . . . . 6 |- ( th -> A e. CC )
5	4	mul01d 8412	. . . . 5 |- ( th -> ( A x. 0 ) = 0 )
6	5	oveq1d 5933	. . . 4 |- ( th -> ( ( A x. 0 ) + ( B x. 1 ) ) = ( 0 + ( B x. 1 ) ) )
7		bezoutlema.b	. . . . . . 7 |- ( th -> B e. NN0 )
8	7	nn0cnd 9295	. . . . . 6 |- ( th -> B e. CC )
9		1cnd 8035	. . . . . 6 |- ( th -> 1 e. CC )
10	8, 9	mulcld 8040	. . . . 5 |- ( th -> ( B x. 1 ) e. CC )
11	10	addlidd 8169	. . . 4 |- ( th -> ( 0 + ( B x. 1 ) ) = ( B x. 1 ) )
12	8	mulridd 8036	. . . 4 |- ( th -> ( B x. 1 ) = B )
13	6, 11, 12	3eqtrrd 2231	. . 3 |- ( th -> B = ( ( A x. 0 ) + ( B x. 1 ) ) )
14		oveq2 5926	. . . . . 6 |- ( s = 0 -> ( A x. s ) = ( A x. 0 ) )
15	14	oveq1d 5933	. . . . 5 |- ( s = 0 -> ( ( A x. s ) + ( B x. t ) ) = ( ( A x. 0 ) + ( B x. t ) ) )
16	15	eqeq2d 2205	. . . 4 |- ( s = 0 -> ( B = ( ( A x. s ) + ( B x. t ) ) <-> B = ( ( A x. 0 ) + ( B x. t ) ) ) )
17		oveq2 5926	. . . . . 6 |- ( t = 1 -> ( B x. t ) = ( B x. 1 ) )
18	17	oveq2d 5934	. . . . 5 |- ( t = 1 -> ( ( A x. 0 ) + ( B x. t ) ) = ( ( A x. 0 ) + ( B x. 1 ) ) )
19	18	eqeq2d 2205	. . . 4 |- ( t = 1 -> ( B = ( ( A x. 0 ) + ( B x. t ) ) <-> B = ( ( A x. 0 ) + ( B x. 1 ) ) ) )
20	16, 19	rspc2ev 2879	. . 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 2200	. . . . . 6 |- ( r = B -> ( r = ( ( A x. s ) + ( B x. t ) ) <-> B = ( ( A x. s ) + ( B x. t ) ) ) )
24	23	2rexbidv 2519	. . . . 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 3018	. . 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 2164   E.wrex 2473   [.wsbc 2985  (class class class)co 5918   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   NN0cn0 9240   ZZcz 9317

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318

This theorem is referenced by:  bezoutlemex  12138