Theorem bezoutlema 11999

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 9278	. . 3 |- 1 e. ZZ
2		0z 9263	. . 3 |- 0 e. ZZ
3		bezoutlema.b	. . . . . . 7 |- ( th -> B e. NN0 )
4	3	nn0cnd 9230	. . . . . 6 |- ( th -> B e. CC )
5	4	mul01d 8349	. . . . 5 |- ( th -> ( B x. 0 ) = 0 )
6	5	oveq2d 5890	. . . 4 |- ( th -> ( ( A x. 1 ) + ( B x. 0 ) ) = ( ( A x. 1 ) + 0 ) )
7		bezoutlema.a	. . . . . . 7 |- ( th -> A e. NN0 )
8	7	nn0cnd 9230	. . . . . 6 |- ( th -> A e. CC )
9		1cnd 7972	. . . . . 6 |- ( th -> 1 e. CC )
10	8, 9	mulcld 7977	. . . . 5 |- ( th -> ( A x. 1 ) e. CC )
11	10	addid1d 8105	. . . 4 |- ( th -> ( ( A x. 1 ) + 0 ) = ( A x. 1 ) )
12	8	mulridd 7973	. . . 4 |- ( th -> ( A x. 1 ) = A )
13	6, 11, 12	3eqtrrd 2215	. . 3 |- ( th -> A = ( ( A x. 1 ) + ( B x. 0 ) ) )
14		oveq2 5882	. . . . . 6 |- ( s = 1 -> ( A x. s ) = ( A x. 1 ) )
15	14	oveq1d 5889	. . . . 5 |- ( s = 1 -> ( ( A x. s ) + ( B x. t ) ) = ( ( A x. 1 ) + ( B x. t ) ) )
16	15	eqeq2d 2189	. . . 4 |- ( s = 1 -> ( A = ( ( A x. s ) + ( B x. t ) ) <-> A = ( ( A x. 1 ) + ( B x. t ) ) ) )
17		oveq2 5882	. . . . . 6 |- ( t = 0 -> ( B x. t ) = ( B x. 0 ) )
18	17	oveq2d 5890	. . . . 5 |- ( t = 0 -> ( ( A x. 1 ) + ( B x. t ) ) = ( ( A x. 1 ) + ( B x. 0 ) ) )
19	18	eqeq2d 2189	. . . 4 |- ( t = 0 -> ( A = ( ( A x. 1 ) + ( B x. t ) ) <-> A = ( ( A x. 1 ) + ( B x. 0 ) ) ) )
20	16, 19	rspc2ev 2856	. . 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 1341	. 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 2184	. . . . . 6 |- ( r = A -> ( r = ( ( A x. s ) + ( B x. t ) ) <-> A = ( ( A x. s ) + ( B x. t ) ) ) )
24	23	2rexbidv 2502	. . . . 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	bitrid 192	. . . 4 |- ( r = A -> ( ph <-> E. s e. ZZ E. t e. ZZ A = ( ( A x. s ) + ( B x. t ) ) ) )
26	25	sbcieg 2995	. . 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 167	1 |- ( th -> [. A / r ]. ph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456   [.wsbc 2962  (class class class)co 5874   0cc0 7810   1c1 7811   + caddc 7813   x. cmul 7815   NN0cn0 9175   ZZcz 9252

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253

This theorem is referenced by:  bezoutlemex  12001