Theorem bezoutlemb 11955

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 9223	. . 3 |- 0 e. ZZ
2		1z 9238	. . 3 |- 1 e. ZZ
3		bezoutlema.a	. . . . . . 7 |- ( th -> A e. NN0 )
4	3	nn0cnd 9190	. . . . . 6 |- ( th -> A e. CC )
5	4	mul01d 8312	. . . . 5 |- ( th -> ( A x. 0 ) = 0 )
6	5	oveq1d 5868	. . . 4 |- ( th -> ( ( A x. 0 ) + ( B x. 1 ) ) = ( 0 + ( B x. 1 ) ) )
7		bezoutlema.b	. . . . . . 7 |- ( th -> B e. NN0 )
8	7	nn0cnd 9190	. . . . . 6 |- ( th -> B e. CC )
9		1cnd 7936	. . . . . 6 |- ( th -> 1 e. CC )
10	8, 9	mulcld 7940	. . . . 5 |- ( th -> ( B x. 1 ) e. CC )
11	10	addid2d 8069	. . . 4 |- ( th -> ( 0 + ( B x. 1 ) ) = ( B x. 1 ) )
12	8	mulid1d 7937	. . . 4 |- ( th -> ( B x. 1 ) = B )
13	6, 11, 12	3eqtrrd 2208	. . 3 |- ( th -> B = ( ( A x. 0 ) + ( B x. 1 ) ) )
14		oveq2 5861	. . . . . 6 |- ( s = 0 -> ( A x. s ) = ( A x. 0 ) )
15	14	oveq1d 5868	. . . . 5 |- ( s = 0 -> ( ( A x. s ) + ( B x. t ) ) = ( ( A x. 0 ) + ( B x. t ) ) )
16	15	eqeq2d 2182	. . . 4 |- ( s = 0 -> ( B = ( ( A x. s ) + ( B x. t ) ) <-> B = ( ( A x. 0 ) + ( B x. t ) ) ) )
17		oveq2 5861	. . . . . 6 |- ( t = 1 -> ( B x. t ) = ( B x. 1 ) )
18	17	oveq2d 5869	. . . . 5 |- ( t = 1 -> ( ( A x. 0 ) + ( B x. t ) ) = ( ( A x. 0 ) + ( B x. 1 ) ) )
19	18	eqeq2d 2182	. . . 4 |- ( t = 1 -> ( B = ( ( A x. 0 ) + ( B x. t ) ) <-> B = ( ( A x. 0 ) + ( B x. 1 ) ) ) )
20	16, 19	rspc2ev 2849	. . 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 1336	. 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 2177	. . . . . 6 |- ( r = B -> ( r = ( ( A x. s ) + ( B x. t ) ) <-> B = ( ( A x. s ) + ( B x. t ) ) ) )
24	23	2rexbidv 2495	. . . . 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 191	. . . 4 |- ( r = B -> ( ph <-> E. s e. ZZ E. t e. ZZ B = ( ( A x. s ) + ( B x. t ) ) ) )
26	25	sbcieg 2987	. . 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 166	1 |- ( th -> [. B / r ]. ph )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   E.wrex 2449   [.wsbc 2955  (class class class)co 5853   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   NN0cn0 9135   ZZcz 9212

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213

This theorem is referenced by:  bezoutlemex  11956