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Theorem recidpirqlemcalc 7629
Description: Lemma for recidpirq 7630. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.)
Hypotheses
Ref Expression
recidpirqlemcalc.a  |-  ( ph  ->  A  e.  P. )
recidpirqlemcalc.b  |-  ( ph  ->  B  e.  P. )
recidpirqlemcalc.rec  |-  ( ph  ->  ( A  .P.  B
)  =  1P )
Assertion
Ref Expression
recidpirqlemcalc  |-  ( ph  ->  ( ( ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  +P.  ( 1P  .P.  1P ) )  +P.  1P )  =  ( ( ( ( A  +P.  1P )  .P.  1P )  +P.  ( 1P  .P.  ( B  +P.  1P ) ) )  +P.  ( 1P 
+P.  1P ) ) )

Proof of Theorem recidpirqlemcalc
StepHypRef Expression
1 recidpirqlemcalc.a . . . . 5  |-  ( ph  ->  A  e.  P. )
2 1pr 7326 . . . . . 6  |-  1P  e.  P.
32a1i 9 . . . . 5  |-  ( ph  ->  1P  e.  P. )
4 addclpr 7309 . . . . 5  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( A  +P.  1P )  e.  P. )
51, 3, 4syl2anc 406 . . . 4  |-  ( ph  ->  ( A  +P.  1P )  e.  P. )
6 recidpirqlemcalc.b . . . . 5  |-  ( ph  ->  B  e.  P. )
7 addclpr 7309 . . . . 5  |-  ( ( B  e.  P.  /\  1P  e.  P. )  -> 
( B  +P.  1P )  e.  P. )
86, 3, 7syl2anc 406 . . . 4  |-  ( ph  ->  ( B  +P.  1P )  e.  P. )
9 addclpr 7309 . . . 4  |-  ( ( ( A  +P.  1P )  e.  P.  /\  ( B  +P.  1P )  e. 
P. )  ->  (
( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P. )
105, 8, 9syl2anc 406 . . 3  |-  ( ph  ->  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P. )
11 addassprg 7351 . . 3  |-  ( ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P.  /\  1P  e.  P.  /\  1P  e.  P. )  ->  ( ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P )  +P. 
1P )  =  ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  ( 1P  +P.  1P ) ) )
1210, 3, 3, 11syl3anc 1199 . 2  |-  ( ph  ->  ( ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P )  +P.  1P )  =  ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  ( 1P  +P.  1P ) ) )
13 distrprg 7360 . . . . . . 7  |-  ( ( ( A  +P.  1P )  e.  P.  /\  B  e.  P.  /\  1P  e.  P. )  ->  ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  =  ( ( ( A  +P.  1P )  .P.  B )  +P.  ( ( A  +P.  1P )  .P. 
1P ) ) )
145, 6, 3, 13syl3anc 1199 . . . . . 6  |-  ( ph  ->  ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  =  ( ( ( A  +P.  1P )  .P.  B )  +P.  ( ( A  +P.  1P )  .P.  1P ) ) )
15 1idpr 7364 . . . . . . . 8  |-  ( ( A  +P.  1P )  e.  P.  ->  (
( A  +P.  1P )  .P.  1P )  =  ( A  +P.  1P ) )
165, 15syl 14 . . . . . . 7  |-  ( ph  ->  ( ( A  +P.  1P )  .P.  1P )  =  ( A  +P.  1P ) )
1716oveq2d 5756 . . . . . 6  |-  ( ph  ->  ( ( ( A  +P.  1P )  .P. 
B )  +P.  (
( A  +P.  1P )  .P.  1P ) )  =  ( ( ( A  +P.  1P )  .P.  B )  +P.  ( A  +P.  1P ) ) )
18 mulcomprg 7352 . . . . . . . . 9  |-  ( ( ( A  +P.  1P )  e.  P.  /\  B  e.  P. )  ->  (
( A  +P.  1P )  .P.  B )  =  ( B  .P.  ( A  +P.  1P ) ) )
195, 6, 18syl2anc 406 . . . . . . . 8  |-  ( ph  ->  ( ( A  +P.  1P )  .P.  B )  =  ( B  .P.  ( A  +P.  1P ) ) )
20 distrprg 7360 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  A  e.  P.  /\  1P  e.  P. )  ->  ( B  .P.  ( A  +P.  1P ) )  =  ( ( B  .P.  A
)  +P.  ( B  .P.  1P ) ) )
216, 1, 3, 20syl3anc 1199 . . . . . . . 8  |-  ( ph  ->  ( B  .P.  ( A  +P.  1P ) )  =  ( ( B  .P.  A )  +P.  ( B  .P.  1P ) ) )
22 mulcomprg 7352 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  .P.  A
)  =  ( A  .P.  B ) )
236, 1, 22syl2anc 406 . . . . . . . . . 10  |-  ( ph  ->  ( B  .P.  A
)  =  ( A  .P.  B ) )
24 recidpirqlemcalc.rec . . . . . . . . . 10  |-  ( ph  ->  ( A  .P.  B
)  =  1P )
2523, 24eqtrd 2148 . . . . . . . . 9  |-  ( ph  ->  ( B  .P.  A
)  =  1P )
26 1idpr 7364 . . . . . . . . . 10  |-  ( B  e.  P.  ->  ( B  .P.  1P )  =  B )
276, 26syl 14 . . . . . . . . 9  |-  ( ph  ->  ( B  .P.  1P )  =  B )
2825, 27oveq12d 5758 . . . . . . . 8  |-  ( ph  ->  ( ( B  .P.  A )  +P.  ( B  .P.  1P ) )  =  ( 1P  +P.  B ) )
2919, 21, 283eqtrd 2152 . . . . . . 7  |-  ( ph  ->  ( ( A  +P.  1P )  .P.  B )  =  ( 1P  +P.  B ) )
3029oveq1d 5755 . . . . . 6  |-  ( ph  ->  ( ( ( A  +P.  1P )  .P. 
B )  +P.  ( A  +P.  1P ) )  =  ( ( 1P 
+P.  B )  +P.  ( A  +P.  1P ) ) )
3114, 17, 303eqtrd 2152 . . . . 5  |-  ( ph  ->  ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  =  ( ( 1P 
+P.  B )  +P.  ( A  +P.  1P ) ) )
32 1idpr 7364 . . . . . 6  |-  ( 1P  e.  P.  ->  ( 1P  .P.  1P )  =  1P )
332, 32mp1i 10 . . . . 5  |-  ( ph  ->  ( 1P  .P.  1P )  =  1P )
3431, 33oveq12d 5758 . . . 4  |-  ( ph  ->  ( ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  +P.  ( 1P  .P.  1P ) )  =  ( ( ( 1P  +P.  B )  +P.  ( A  +P.  1P ) )  +P.  1P ) )
35 addcomprg 7350 . . . . . . . 8  |-  ( ( 1P  e.  P.  /\  B  e.  P. )  ->  ( 1P  +P.  B
)  =  ( B  +P.  1P ) )
363, 6, 35syl2anc 406 . . . . . . 7  |-  ( ph  ->  ( 1P  +P.  B
)  =  ( B  +P.  1P ) )
3736oveq1d 5755 . . . . . 6  |-  ( ph  ->  ( ( 1P  +P.  B )  +P.  ( A  +P.  1P ) )  =  ( ( B  +P.  1P )  +P.  ( A  +P.  1P ) ) )
38 addcomprg 7350 . . . . . . 7  |-  ( ( ( B  +P.  1P )  e.  P.  /\  ( A  +P.  1P )  e. 
P. )  ->  (
( B  +P.  1P )  +P.  ( A  +P.  1P ) )  =  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) ) )
398, 5, 38syl2anc 406 . . . . . 6  |-  ( ph  ->  ( ( B  +P.  1P )  +P.  ( A  +P.  1P ) )  =  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) ) )
4037, 39eqtrd 2148 . . . . 5  |-  ( ph  ->  ( ( 1P  +P.  B )  +P.  ( A  +P.  1P ) )  =  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) ) )
4140oveq1d 5755 . . . 4  |-  ( ph  ->  ( ( ( 1P 
+P.  B )  +P.  ( A  +P.  1P ) )  +P.  1P )  =  ( (
( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P ) )
4234, 41eqtrd 2148 . . 3  |-  ( ph  ->  ( ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  +P.  ( 1P  .P.  1P ) )  =  ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P ) )
4342oveq1d 5755 . 2  |-  ( ph  ->  ( ( ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  +P.  ( 1P  .P.  1P ) )  +P.  1P )  =  ( ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P )  +P.  1P ) )
44 mulcomprg 7352 . . . . . 6  |-  ( ( 1P  e.  P.  /\  ( B  +P.  1P )  e.  P. )  -> 
( 1P  .P.  ( B  +P.  1P ) )  =  ( ( B  +P.  1P )  .P. 
1P ) )
453, 8, 44syl2anc 406 . . . . 5  |-  ( ph  ->  ( 1P  .P.  ( B  +P.  1P ) )  =  ( ( B  +P.  1P )  .P. 
1P ) )
46 1idpr 7364 . . . . . 6  |-  ( ( B  +P.  1P )  e.  P.  ->  (
( B  +P.  1P )  .P.  1P )  =  ( B  +P.  1P ) )
478, 46syl 14 . . . . 5  |-  ( ph  ->  ( ( B  +P.  1P )  .P.  1P )  =  ( B  +P.  1P ) )
4845, 47eqtrd 2148 . . . 4  |-  ( ph  ->  ( 1P  .P.  ( B  +P.  1P ) )  =  ( B  +P.  1P ) )
4916, 48oveq12d 5758 . . 3  |-  ( ph  ->  ( ( ( A  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( B  +P.  1P ) ) )  =  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) ) )
5049oveq1d 5755 . 2  |-  ( ph  ->  ( ( ( ( A  +P.  1P )  .P.  1P )  +P.  ( 1P  .P.  ( B  +P.  1P ) ) )  +P.  ( 1P 
+P.  1P ) )  =  ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  ( 1P  +P.  1P ) ) )
5112, 43, 503eqtr4d 2158 1  |-  ( ph  ->  ( ( ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  +P.  ( 1P  .P.  1P ) )  +P.  1P )  =  ( ( ( ( A  +P.  1P )  .P.  1P )  +P.  ( 1P  .P.  ( B  +P.  1P ) ) )  +P.  ( 1P 
+P.  1P ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463  (class class class)co 5740   P.cnp 7063   1Pc1p 7064    +P. cpp 7065    .P. cmp 7066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-i1p 7239  df-iplp 7240  df-imp 7241
This theorem is referenced by:  recidpirq  7630
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