Theorem bezoutlema 12754

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 9649	. . 3 |- 1 e. ZZ
2		0z 9634	. . 3 |- 0 e. ZZ
3		bezoutlema.b	. . . . . . 7 |- ( th -> B e. NN0 )
4	3	nn0cnd 9601	. . . . . 6 |- ( th -> B e. CC )
5	4	mul01d 8710	. . . . 5 |- ( th -> ( B x. 0 ) = 0 )
6	5	oveq2d 6091	. . . 4 |- ( th -> ( ( A x. 1 ) + ( B x. 0 ) ) = ( ( A x. 1 ) + 0 ) )
7		bezoutlema.a	. . . . . . 7 |- ( th -> A e. NN0 )
8	7	nn0cnd 9601	. . . . . 6 |- ( th -> A e. CC )
9		1cnd 8332	. . . . . 6 |- ( th -> 1 e. CC )
10	8, 9	mulcld 8336	. . . . 5 |- ( th -> ( A x. 1 ) e. CC )
11	10	addridd 8465	. . . 4 |- ( th -> ( ( A x. 1 ) + 0 ) = ( A x. 1 ) )
12	8	mulridd 8333	. . . 4 |- ( th -> ( A x. 1 ) = A )
13	6, 11, 12	3eqtrrd 2276	. . 3 |- ( th -> A = ( ( A x. 1 ) + ( B x. 0 ) ) )
14		oveq2 6083	. . . . . 6 |- ( s = 1 -> ( A x. s ) = ( A x. 1 ) )
15	14	oveq1d 6090	. . . . 5 |- ( s = 1 -> ( ( A x. s ) + ( B x. t ) ) = ( ( A x. 1 ) + ( B x. t ) ) )
16	15	eqeq2d 2250	. . . 4 |- ( s = 1 -> ( A = ( ( A x. s ) + ( B x. t ) ) <-> A = ( ( A x. 1 ) + ( B x. t ) ) ) )
17		oveq2 6083	. . . . . 6 |- ( t = 0 -> ( B x. t ) = ( B x. 0 ) )
18	17	oveq2d 6091	. . . . 5 |- ( t = 0 -> ( ( A x. 1 ) + ( B x. t ) ) = ( ( A x. 1 ) + ( B x. 0 ) ) )
19	18	eqeq2d 2250	. . . 4 |- ( t = 0 -> ( A = ( ( A x. 1 ) + ( B x. t ) ) <-> A = ( ( A x. 1 ) + ( B x. 0 ) ) ) )
20	16, 19	rspc2ev 2945	. . 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 1382	. 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 2245	. . . . . 6 |- ( r = A -> ( r = ( ( A x. s ) + ( B x. t ) ) <-> A = ( ( A x. s ) + ( B x. t ) ) ) )
24	23	2rexbidv 2575	. . . . 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 3084	. . 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 1402   e. wcel 2209   E.wrex 2529   [.wsbc 3051  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   NN0cn0 9542   ZZcz 9623

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624

This theorem is referenced by:  bezoutlemex  12756