Theorem bezoutlema 12435

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 9433	. . 3 |- 1 e. ZZ
2		0z 9418	. . 3 |- 0 e. ZZ
3		bezoutlema.b	. . . . . . 7 |- ( th -> B e. NN0 )
4	3	nn0cnd 9385	. . . . . 6 |- ( th -> B e. CC )
5	4	mul01d 8500	. . . . 5 |- ( th -> ( B x. 0 ) = 0 )
6	5	oveq2d 5983	. . . 4 |- ( th -> ( ( A x. 1 ) + ( B x. 0 ) ) = ( ( A x. 1 ) + 0 ) )
7		bezoutlema.a	. . . . . . 7 |- ( th -> A e. NN0 )
8	7	nn0cnd 9385	. . . . . 6 |- ( th -> A e. CC )
9		1cnd 8123	. . . . . 6 |- ( th -> 1 e. CC )
10	8, 9	mulcld 8128	. . . . 5 |- ( th -> ( A x. 1 ) e. CC )
11	10	addridd 8256	. . . 4 |- ( th -> ( ( A x. 1 ) + 0 ) = ( A x. 1 ) )
12	8	mulridd 8124	. . . 4 |- ( th -> ( A x. 1 ) = A )
13	6, 11, 12	3eqtrrd 2245	. . 3 |- ( th -> A = ( ( A x. 1 ) + ( B x. 0 ) ) )
14		oveq2 5975	. . . . . 6 |- ( s = 1 -> ( A x. s ) = ( A x. 1 ) )
15	14	oveq1d 5982	. . . . 5 |- ( s = 1 -> ( ( A x. s ) + ( B x. t ) ) = ( ( A x. 1 ) + ( B x. t ) ) )
16	15	eqeq2d 2219	. . . 4 |- ( s = 1 -> ( A = ( ( A x. s ) + ( B x. t ) ) <-> A = ( ( A x. 1 ) + ( B x. t ) ) ) )
17		oveq2 5975	. . . . . 6 |- ( t = 0 -> ( B x. t ) = ( B x. 0 ) )
18	17	oveq2d 5983	. . . . 5 |- ( t = 0 -> ( ( A x. 1 ) + ( B x. t ) ) = ( ( A x. 1 ) + ( B x. 0 ) ) )
19	18	eqeq2d 2219	. . . 4 |- ( t = 0 -> ( A = ( ( A x. 1 ) + ( B x. t ) ) <-> A = ( ( A x. 1 ) + ( B x. 0 ) ) ) )
20	16, 19	rspc2ev 2899	. . 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 1354	. 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 2214	. . . . . 6 |- ( r = A -> ( r = ( ( A x. s ) + ( B x. t ) ) <-> A = ( ( A x. s ) + ( B x. t ) ) ) )
24	23	2rexbidv 2533	. . . . 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 3038	. . 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 1373   e. wcel 2178   E.wrex 2487   [.wsbc 3005  (class class class)co 5967   0cc0 7960   1c1 7961   + caddc 7963   x. cmul 7965   NN0cn0 9330   ZZcz 9407

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408

This theorem is referenced by:  bezoutlemex  12437