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Theorem recidpirqlemcalc 7859
Description: Lemma for recidpirq 7860. 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 7556 . . . . . 6  |-  1P  e.  P.
32a1i 9 . . . . 5  |-  ( ph  ->  1P  e.  P. )
4 addclpr 7539 . . . . 5  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( A  +P.  1P )  e.  P. )
51, 3, 4syl2anc 411 . . . 4  |-  ( ph  ->  ( A  +P.  1P )  e.  P. )
6 recidpirqlemcalc.b . . . . 5  |-  ( ph  ->  B  e.  P. )
7 addclpr 7539 . . . . 5  |-  ( ( B  e.  P.  /\  1P  e.  P. )  -> 
( B  +P.  1P )  e.  P. )
86, 3, 7syl2anc 411 . . . 4  |-  ( ph  ->  ( B  +P.  1P )  e.  P. )
9 addclpr 7539 . . . 4  |-  ( ( ( A  +P.  1P )  e.  P.  /\  ( B  +P.  1P )  e. 
P. )  ->  (
( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P. )
105, 8, 9syl2anc 411 . . 3  |-  ( ph  ->  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  e.  P. )
11 addassprg 7581 . . 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 1238 . 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 7590 . . . . . . 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 1238 . . . . . 6  |-  ( ph  ->  ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  =  ( ( ( A  +P.  1P )  .P.  B )  +P.  ( ( A  +P.  1P )  .P.  1P ) ) )
15 1idpr 7594 . . . . . . . 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 5894 . . . . . 6  |-  ( ph  ->  ( ( ( A  +P.  1P )  .P. 
B )  +P.  (
( A  +P.  1P )  .P.  1P ) )  =  ( ( ( A  +P.  1P )  .P.  B )  +P.  ( A  +P.  1P ) ) )
18 mulcomprg 7582 . . . . . . . . 9  |-  ( ( ( A  +P.  1P )  e.  P.  /\  B  e.  P. )  ->  (
( A  +P.  1P )  .P.  B )  =  ( B  .P.  ( A  +P.  1P ) ) )
195, 6, 18syl2anc 411 . . . . . . . 8  |-  ( ph  ->  ( ( A  +P.  1P )  .P.  B )  =  ( B  .P.  ( A  +P.  1P ) ) )
20 distrprg 7590 . . . . . . . . 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 1238 . . . . . . . 8  |-  ( ph  ->  ( B  .P.  ( A  +P.  1P ) )  =  ( ( B  .P.  A )  +P.  ( B  .P.  1P ) ) )
22 mulcomprg 7582 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( B  .P.  A
)  =  ( A  .P.  B ) )
236, 1, 22syl2anc 411 . . . . . . . . . 10  |-  ( ph  ->  ( B  .P.  A
)  =  ( A  .P.  B ) )
24 recidpirqlemcalc.rec . . . . . . . . . 10  |-  ( ph  ->  ( A  .P.  B
)  =  1P )
2523, 24eqtrd 2210 . . . . . . . . 9  |-  ( ph  ->  ( B  .P.  A
)  =  1P )
26 1idpr 7594 . . . . . . . . . 10  |-  ( B  e.  P.  ->  ( B  .P.  1P )  =  B )
276, 26syl 14 . . . . . . . . 9  |-  ( ph  ->  ( B  .P.  1P )  =  B )
2825, 27oveq12d 5896 . . . . . . . 8  |-  ( ph  ->  ( ( B  .P.  A )  +P.  ( B  .P.  1P ) )  =  ( 1P  +P.  B ) )
2919, 21, 283eqtrd 2214 . . . . . . 7  |-  ( ph  ->  ( ( A  +P.  1P )  .P.  B )  =  ( 1P  +P.  B ) )
3029oveq1d 5893 . . . . . 6  |-  ( ph  ->  ( ( ( A  +P.  1P )  .P. 
B )  +P.  ( A  +P.  1P ) )  =  ( ( 1P 
+P.  B )  +P.  ( A  +P.  1P ) ) )
3114, 17, 303eqtrd 2214 . . . . 5  |-  ( ph  ->  ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  =  ( ( 1P 
+P.  B )  +P.  ( A  +P.  1P ) ) )
32 1idpr 7594 . . . . . 6  |-  ( 1P  e.  P.  ->  ( 1P  .P.  1P )  =  1P )
332, 32mp1i 10 . . . . 5  |-  ( ph  ->  ( 1P  .P.  1P )  =  1P )
3431, 33oveq12d 5896 . . . 4  |-  ( ph  ->  ( ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  +P.  ( 1P  .P.  1P ) )  =  ( ( ( 1P  +P.  B )  +P.  ( A  +P.  1P ) )  +P.  1P ) )
35 addcomprg 7580 . . . . . . . 8  |-  ( ( 1P  e.  P.  /\  B  e.  P. )  ->  ( 1P  +P.  B
)  =  ( B  +P.  1P ) )
363, 6, 35syl2anc 411 . . . . . . 7  |-  ( ph  ->  ( 1P  +P.  B
)  =  ( B  +P.  1P ) )
3736oveq1d 5893 . . . . . 6  |-  ( ph  ->  ( ( 1P  +P.  B )  +P.  ( A  +P.  1P ) )  =  ( ( B  +P.  1P )  +P.  ( A  +P.  1P ) ) )
38 addcomprg 7580 . . . . . . 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 411 . . . . . 6  |-  ( ph  ->  ( ( B  +P.  1P )  +P.  ( A  +P.  1P ) )  =  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) ) )
4037, 39eqtrd 2210 . . . . 5  |-  ( ph  ->  ( ( 1P  +P.  B )  +P.  ( A  +P.  1P ) )  =  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) ) )
4140oveq1d 5893 . . . 4  |-  ( ph  ->  ( ( ( 1P 
+P.  B )  +P.  ( A  +P.  1P ) )  +P.  1P )  =  ( (
( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P ) )
4234, 41eqtrd 2210 . . 3  |-  ( ph  ->  ( ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  +P.  ( 1P  .P.  1P ) )  =  ( ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) )  +P.  1P ) )
4342oveq1d 5893 . 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 7582 . . . . . 6  |-  ( ( 1P  e.  P.  /\  ( B  +P.  1P )  e.  P. )  -> 
( 1P  .P.  ( B  +P.  1P ) )  =  ( ( B  +P.  1P )  .P. 
1P ) )
453, 8, 44syl2anc 411 . . . . 5  |-  ( ph  ->  ( 1P  .P.  ( B  +P.  1P ) )  =  ( ( B  +P.  1P )  .P. 
1P ) )
46 1idpr 7594 . . . . . 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 2210 . . . 4  |-  ( ph  ->  ( 1P  .P.  ( B  +P.  1P ) )  =  ( B  +P.  1P ) )
4916, 48oveq12d 5896 . . 3  |-  ( ph  ->  ( ( ( A  +P.  1P )  .P. 
1P )  +P.  ( 1P  .P.  ( B  +P.  1P ) ) )  =  ( ( A  +P.  1P )  +P.  ( B  +P.  1P ) ) )
5049oveq1d 5893 . 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 2220 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 1353    e. wcel 2148  (class class class)co 5878   P.cnp 7293   1Pc1p 7294    +P. cpp 7295    .P. cmp 7296
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-1o 6420  df-2o 6421  df-oadd 6424  df-omul 6425  df-er 6538  df-ec 6540  df-qs 6544  df-ni 7306  df-pli 7307  df-mi 7308  df-lti 7309  df-plpq 7346  df-mpq 7347  df-enq 7349  df-nqqs 7350  df-plqqs 7351  df-mqqs 7352  df-1nqqs 7353  df-rq 7354  df-ltnqqs 7355  df-enq0 7426  df-nq0 7427  df-0nq0 7428  df-plq0 7429  df-mq0 7430  df-inp 7468  df-i1p 7469  df-iplp 7470  df-imp 7471
This theorem is referenced by:  recidpirq  7860
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