Theorem recidpirqlemcalc 7448

Description: Lemma for recidpirq 7449. 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

Step	Hyp	Ref	Expression
1		recidpirqlemcalc.a	. . . . 5 |- ( ph -> A e. P. )
2		1pr 7167	. . . . . 6 |- 1P e. P.
3	2	a1i 9	. . . . 5 |- ( ph -> 1P e. P. )
4		addclpr 7150	. . . . 5 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
5	1, 3, 4	syl2anc 404	. . . 4 |- ( ph -> ( A +P. 1P ) e. P. )
6		recidpirqlemcalc.b	. . . . 5 |- ( ph -> B e. P. )
7		addclpr 7150	. . . . 5 |- ( ( B e. P. /\ 1P e. P. ) -> ( B +P. 1P ) e. P. )
8	6, 3, 7	syl2anc 404	. . . 4 |- ( ph -> ( B +P. 1P ) e. P. )
9		addclpr 7150	. . . 4 |- ( ( ( A +P. 1P ) e. P. /\ ( B +P. 1P ) e. P. ) -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) e. P. )
10	5, 8, 9	syl2anc 404	. . 3 |- ( ph -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) e. P. )
11		addassprg 7192	. . 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 ) ) )
12	10, 3, 3, 11	syl3anc 1175	. 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 7201	. . . . . . 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 ) ) )
14	5, 6, 3, 13	syl3anc 1175	. . . . . 6 |- ( ph -> ( ( A +P. 1P ) .P. ( B +P. 1P ) ) = ( ( ( A +P. 1P ) .P. B ) +P. ( ( A +P. 1P ) .P. 1P ) ) )
15		1idpr 7205	. . . . . . . 8 |- ( ( A +P. 1P ) e. P. -> ( ( A +P. 1P ) .P. 1P ) = ( A +P. 1P ) )
16	5, 15	syl 14	. . . . . . 7 |- ( ph -> ( ( A +P. 1P ) .P. 1P ) = ( A +P. 1P ) )
17	16	oveq2d 5682	. . . . . 6 |- ( ph -> ( ( ( A +P. 1P ) .P. B ) +P. ( ( A +P. 1P ) .P. 1P ) ) = ( ( ( A +P. 1P ) .P. B ) +P. ( A +P. 1P ) ) )
18		mulcomprg 7193	. . . . . . . . 9 |- ( ( ( A +P. 1P ) e. P. /\ B e. P. ) -> ( ( A +P. 1P ) .P. B ) = ( B .P. ( A +P. 1P ) ) )
19	5, 6, 18	syl2anc 404	. . . . . . . 8 |- ( ph -> ( ( A +P. 1P ) .P. B ) = ( B .P. ( A +P. 1P ) ) )
20		distrprg 7201	. . . . . . . . 9 |- ( ( B e. P. /\ A e. P. /\ 1P e. P. ) -> ( B .P. ( A +P. 1P ) ) = ( ( B .P. A ) +P. ( B .P. 1P ) ) )
21	6, 1, 3, 20	syl3anc 1175	. . . . . . . 8 |- ( ph -> ( B .P. ( A +P. 1P ) ) = ( ( B .P. A ) +P. ( B .P. 1P ) ) )
22		mulcomprg 7193	. . . . . . . . . . 11 |- ( ( B e. P. /\ A e. P. ) -> ( B .P. A ) = ( A .P. B ) )
23	6, 1, 22	syl2anc 404	. . . . . . . . . 10 |- ( ph -> ( B .P. A ) = ( A .P. B ) )
24		recidpirqlemcalc.rec	. . . . . . . . . 10 |- ( ph -> ( A .P. B ) = 1P )
25	23, 24	eqtrd 2121	. . . . . . . . 9 |- ( ph -> ( B .P. A ) = 1P )
26		1idpr 7205	. . . . . . . . . 10 |- ( B e. P. -> ( B .P. 1P ) = B )
27	6, 26	syl 14	. . . . . . . . 9 |- ( ph -> ( B .P. 1P ) = B )
28	25, 27	oveq12d 5684	. . . . . . . 8 |- ( ph -> ( ( B .P. A ) +P. ( B .P. 1P ) ) = ( 1P +P. B ) )
29	19, 21, 28	3eqtrd 2125	. . . . . . 7 |- ( ph -> ( ( A +P. 1P ) .P. B ) = ( 1P +P. B ) )
30	29	oveq1d 5681	. . . . . 6 |- ( ph -> ( ( ( A +P. 1P ) .P. B ) +P. ( A +P. 1P ) ) = ( ( 1P +P. B ) +P. ( A +P. 1P ) ) )
31	14, 17, 30	3eqtrd 2125	. . . . 5 |- ( ph -> ( ( A +P. 1P ) .P. ( B +P. 1P ) ) = ( ( 1P +P. B ) +P. ( A +P. 1P ) ) )
32		1idpr 7205	. . . . . 6 |- ( 1P e. P. -> ( 1P .P. 1P ) = 1P )
33	2, 32	mp1i 10	. . . . 5 |- ( ph -> ( 1P .P. 1P ) = 1P )
34	31, 33	oveq12d 5684	. . . 4 |- ( ph -> ( ( ( A +P. 1P ) .P. ( B +P. 1P ) ) +P. ( 1P .P. 1P ) ) = ( ( ( 1P +P. B ) +P. ( A +P. 1P ) ) +P. 1P ) )
35		addcomprg 7191	. . . . . . . 8 |- ( ( 1P e. P. /\ B e. P. ) -> ( 1P +P. B ) = ( B +P. 1P ) )
36	3, 6, 35	syl2anc 404	. . . . . . 7 |- ( ph -> ( 1P +P. B ) = ( B +P. 1P ) )
37	36	oveq1d 5681	. . . . . 6 |- ( ph -> ( ( 1P +P. B ) +P. ( A +P. 1P ) ) = ( ( B +P. 1P ) +P. ( A +P. 1P ) ) )
38		addcomprg 7191	. . . . . . 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 ) ) )
39	8, 5, 38	syl2anc 404	. . . . . 6 |- ( ph -> ( ( B +P. 1P ) +P. ( A +P. 1P ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
40	37, 39	eqtrd 2121	. . . . 5 |- ( ph -> ( ( 1P +P. B ) +P. ( A +P. 1P ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
41	40	oveq1d 5681	. . . 4 |- ( ph -> ( ( ( 1P +P. B ) +P. ( A +P. 1P ) ) +P. 1P ) = ( ( ( A +P. 1P ) +P. ( B +P. 1P ) ) +P. 1P ) )
42	34, 41	eqtrd 2121	. . 3 |- ( ph -> ( ( ( A +P. 1P ) .P. ( B +P. 1P ) ) +P. ( 1P .P. 1P ) ) = ( ( ( A +P. 1P ) +P. ( B +P. 1P ) ) +P. 1P ) )
43	42	oveq1d 5681	. 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 7193	. . . . . 6 |- ( ( 1P e. P. /\ ( B +P. 1P ) e. P. ) -> ( 1P .P. ( B +P. 1P ) ) = ( ( B +P. 1P ) .P. 1P ) )
45	3, 8, 44	syl2anc 404	. . . . 5 |- ( ph -> ( 1P .P. ( B +P. 1P ) ) = ( ( B +P. 1P ) .P. 1P ) )
46		1idpr 7205	. . . . . 6 |- ( ( B +P. 1P ) e. P. -> ( ( B +P. 1P ) .P. 1P ) = ( B +P. 1P ) )
47	8, 46	syl 14	. . . . 5 |- ( ph -> ( ( B +P. 1P ) .P. 1P ) = ( B +P. 1P ) )
48	45, 47	eqtrd 2121	. . . 4 |- ( ph -> ( 1P .P. ( B +P. 1P ) ) = ( B +P. 1P ) )
49	16, 48	oveq12d 5684	. . 3 |- ( ph -> ( ( ( A +P. 1P ) .P. 1P ) +P. ( 1P .P. ( B +P. 1P ) ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
50	49	oveq1d 5681	. 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 ) ) )
51	12, 43, 50	3eqtr4d 2131	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 ) ) )

Syntax hints:   -> wi 4   = wceq 1290   e. wcel 1439  (class class class)co 5666   P.cnp 6904   1Pc1p 6905   +P. cpp 6906   .P. cmp 6907

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-2o 6196  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6917  df-pli 6918  df-mi 6919  df-lti 6920  df-plpq 6957  df-mpq 6958  df-enq 6960  df-nqqs 6961  df-plqqs 6962  df-mqqs 6963  df-1nqqs 6964  df-rq 6965  df-ltnqqs 6966  df-enq0 7037  df-nq0 7038  df-0nq0 7039  df-plq0 7040  df-mq0 7041  df-inp 7079  df-i1p 7080  df-iplp 7081  df-imp 7082

This theorem is referenced by:  recidpirq  7449