Theorem divalglemqt 12605

Description: Lemma for divalg 12610. The Q = T case involved in showing uniqueness. (Contributed by Jim Kingdon, 5-Dec-2021.)

Hypotheses

Ref	Expression
divalglemqt.d	|- ( ph -> D e. ZZ )
divalglemqt.r	|- ( ph -> R e. ZZ )
divalglemqt.s	|- ( ph -> S e. ZZ )
divalglemqt.q	|- ( ph -> Q e. ZZ )
divalglemqt.t	|- ( ph -> T e. ZZ )
divalglemqt.qt	|- ( ph -> Q = T )
divalglemqt.eq	|- ( ph -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) )

Assertion

Ref	Expression
divalglemqt	|- ( ph -> R = S )

Proof of Theorem divalglemqt

Step	Hyp	Ref	Expression
1		divalglemqt.qt	. . . 4 |- ( ph -> Q = T )
2	1	oveq1d 6065	. . 3 |- ( ph -> ( Q x. D ) = ( T x. D ) )
3		divalglemqt.q	. . . . 5 |- ( ph -> Q e. ZZ )
4		divalglemqt.d	. . . . 5 |- ( ph -> D e. ZZ )
5	3, 4	zmulcld 9706	. . . 4 |- ( ph -> ( Q x. D ) e. ZZ )
6	5	zcnd 9701	. . 3 |- ( ph -> ( Q x. D ) e. CC )
7	2, 6	eqeltrrd 2310	. 2 |- ( ph -> ( T x. D ) e. CC )
8		divalglemqt.r	. . 3 |- ( ph -> R e. ZZ )
9	8	zcnd 9701	. 2 |- ( ph -> R e. CC )
10		divalglemqt.s	. . 3 |- ( ph -> S e. ZZ )
11	10	zcnd 9701	. 2 |- ( ph -> S e. CC )
12	2	oveq1d 6065	. . 3 |- ( ph -> ( ( Q x. D ) + R ) = ( ( T x. D ) + R ) )
13		divalglemqt.eq	. . 3 |- ( ph -> ( ( Q x. D ) + R ) = ( ( T x. D ) + S ) )
14	12, 13	eqtr3d 2267	. 2 |- ( ph -> ( ( T x. D ) + R ) = ( ( T x. D ) + S ) )
15	7, 9, 11, 14	addcanad 8459	1 |- ( ph -> R = S )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203  (class class class)co 6050   CCcc 8125   + caddc 8130   x. cmul 8132   ZZcz 9577

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578

This theorem is referenced by:  divalglemeunn  12607  divalglemeuneg  12609