Theorem divalglemqt 12263

Description: Lemma for divalg 12268. 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 5961	. . 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 9503	. . . 4 |- ( ph -> ( Q x. D ) e. ZZ )
6	5	zcnd 9498	. . 3 |- ( ph -> ( Q x. D ) e. CC )
7	2, 6	eqeltrrd 2283	. 2 |- ( ph -> ( T x. D ) e. CC )
8		divalglemqt.r	. . 3 |- ( ph -> R e. ZZ )
9	8	zcnd 9498	. 2 |- ( ph -> R e. CC )
10		divalglemqt.s	. . 3 |- ( ph -> S e. ZZ )
11	10	zcnd 9498	. 2 |- ( ph -> S e. CC )
12	2	oveq1d 5961	. . 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 2240	. 2 |- ( ph -> ( ( T x. D ) + R ) = ( ( T x. D ) + S ) )
15	7, 9, 11, 14	addcanad 8260	1 |- ( ph -> R = S )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176  (class class class)co 5946   CCcc 7925   + caddc 7930   x. cmul 7932   ZZcz 9374

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-sub 8247  df-neg 8248  df-inn 9039  df-n0 9298  df-z 9375

This theorem is referenced by:  divalglemeunn  12265  divalglemeuneg  12267