Theorem bezoutlemb 12572

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 9490	. . 3 |- 0 e. ZZ
2		1z 9505	. . 3 |- 1 e. ZZ
3		bezoutlema.a	. . . . . . 7 |- ( th -> A e. NN0 )
4	3	nn0cnd 9457	. . . . . 6 |- ( th -> A e. CC )
5	4	mul01d 8572	. . . . 5 |- ( th -> ( A x. 0 ) = 0 )
6	5	oveq1d 6033	. . . 4 |- ( th -> ( ( A x. 0 ) + ( B x. 1 ) ) = ( 0 + ( B x. 1 ) ) )
7		bezoutlema.b	. . . . . . 7 |- ( th -> B e. NN0 )
8	7	nn0cnd 9457	. . . . . 6 |- ( th -> B e. CC )
9		1cnd 8195	. . . . . 6 |- ( th -> 1 e. CC )
10	8, 9	mulcld 8200	. . . . 5 |- ( th -> ( B x. 1 ) e. CC )
11	10	addlidd 8329	. . . 4 |- ( th -> ( 0 + ( B x. 1 ) ) = ( B x. 1 ) )
12	8	mulridd 8196	. . . 4 |- ( th -> ( B x. 1 ) = B )
13	6, 11, 12	3eqtrrd 2269	. . 3 |- ( th -> B = ( ( A x. 0 ) + ( B x. 1 ) ) )
14		oveq2 6026	. . . . . 6 |- ( s = 0 -> ( A x. s ) = ( A x. 0 ) )
15	14	oveq1d 6033	. . . . 5 |- ( s = 0 -> ( ( A x. s ) + ( B x. t ) ) = ( ( A x. 0 ) + ( B x. t ) ) )
16	15	eqeq2d 2243	. . . 4 |- ( s = 0 -> ( B = ( ( A x. s ) + ( B x. t ) ) <-> B = ( ( A x. 0 ) + ( B x. t ) ) ) )
17		oveq2 6026	. . . . . 6 |- ( t = 1 -> ( B x. t ) = ( B x. 1 ) )
18	17	oveq2d 6034	. . . . 5 |- ( t = 1 -> ( ( A x. 0 ) + ( B x. t ) ) = ( ( A x. 0 ) + ( B x. 1 ) ) )
19	18	eqeq2d 2243	. . . 4 |- ( t = 1 -> ( B = ( ( A x. 0 ) + ( B x. t ) ) <-> B = ( ( A x. 0 ) + ( B x. 1 ) ) ) )
20	16, 19	rspc2ev 2925	. . 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 1377	. 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 2238	. . . . . 6 |- ( r = B -> ( r = ( ( A x. s ) + ( B x. t ) ) <-> B = ( ( A x. s ) + ( B x. t ) ) ) )
24	23	2rexbidv 2557	. . . . 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 3064	. . 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 1397   e. wcel 2202   E.wrex 2511   [.wsbc 3031  (class class class)co 6018   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   NN0cn0 9402   ZZcz 9479

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480

This theorem is referenced by:  bezoutlemex  12573