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Theorem congtr 27030
 Description: A wff of the form is interpreted as a congruential equation. This is similar to , but is defined such that behavior is regular for zero and negative values of . To use this concept effectively, we need to show that congruential equations behave similarly to normal equations; first a transitivity law. Idea for the future: If there was a congruential equation symbol, it could incorporate type constraints, so that most of these would not need them. (Contributed by Stefan O'Rear, 1-Oct-2014.)
Assertion
Ref Expression
congtr

Proof of Theorem congtr
StepHypRef Expression
1 simp1l 981 . . 3
2 simp1r 982 . . . 4
3 simp2l 983 . . . 4
42, 3zsubcld 10380 . . 3
5 zsubcl 10319 . . . 4
653ad2ant2 979 . . 3
7 simp3 959 . . 3
8 dvds2add 12881 . . . 4
98imp 419 . . 3
101, 4, 6, 7, 9syl31anc 1187 . 2
11 zcn 10287 . . . . 5
1211adantl 453 . . . 4
13123ad2ant1 978 . . 3
14 zcn 10287 . . . . 5
1514adantr 452 . . . 4
16153ad2ant2 979 . . 3
17 zcn 10287 . . . . 5
1817adantl 453 . . . 4
19183ad2ant2 979 . . 3
2013, 16, 19npncand 9435 . 2
2110, 20breqtrd 4236 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wcel 1725   class class class wbr 4212  (class class class)co 6081  cc 8988   caddc 8993   cmin 9291  cz 10282   cdivides 12852 This theorem is referenced by:  congmul  27032  acongtr  27043  jm2.18  27059  jm2.27a  27076 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-dvds 12853
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