Theorem divalglemqt 12479

Description: Lemma for divalg 12484. 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 6032	. . 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 9607	. . . 4 |- ( ph -> ( Q x. D ) e. ZZ )
6	5	zcnd 9602	. . 3 |- ( ph -> ( Q x. D ) e. CC )
7	2, 6	eqeltrrd 2309	. 2 |- ( ph -> ( T x. D ) e. CC )
8		divalglemqt.r	. . 3 |- ( ph -> R e. ZZ )
9	8	zcnd 9602	. 2 |- ( ph -> R e. CC )
10		divalglemqt.s	. . 3 |- ( ph -> S e. ZZ )
11	10	zcnd 9602	. 2 |- ( ph -> S e. CC )
12	2	oveq1d 6032	. . 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 2266	. 2 |- ( ph -> ( ( T x. D ) + R ) = ( ( T x. D ) + S ) )
15	7, 9, 11, 14	addcanad 8364	1 |- ( ph -> R = S )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202  (class class class)co 6017   CCcc 8029   + caddc 8034   x. cmul 8036   ZZcz 9478

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-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

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-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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479

This theorem is referenced by:  divalglemeunn  12481  divalglemeuneg  12483