Theorem recidpirqlemcalc 7869

Description: Lemma for recidpirq 7870. 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 7566	. . . . . 6 |- 1P e. P.
3	2	a1i 9	. . . . 5 |- ( ph -> 1P e. P. )
4		addclpr 7549	. . . . 5 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
5	1, 3, 4	syl2anc 411	. . . 4 |- ( ph -> ( A +P. 1P ) e. P. )
6		recidpirqlemcalc.b	. . . . 5 |- ( ph -> B e. P. )
7		addclpr 7549	. . . . 5 |- ( ( B e. P. /\ 1P e. P. ) -> ( B +P. 1P ) e. P. )
8	6, 3, 7	syl2anc 411	. . . 4 |- ( ph -> ( B +P. 1P ) e. P. )
9		addclpr 7549	. . . 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 411	. . 3 |- ( ph -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) e. P. )
11		addassprg 7591	. . 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 1248	. 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 7600	. . . . . . 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 1248	. . . . . 6 |- ( ph -> ( ( A +P. 1P ) .P. ( B +P. 1P ) ) = ( ( ( A +P. 1P ) .P. B ) +P. ( ( A +P. 1P ) .P. 1P ) ) )
15		1idpr 7604	. . . . . . . 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 5904	. . . . . 6 |- ( ph -> ( ( ( A +P. 1P ) .P. B ) +P. ( ( A +P. 1P ) .P. 1P ) ) = ( ( ( A +P. 1P ) .P. B ) +P. ( A +P. 1P ) ) )
18		mulcomprg 7592	. . . . . . . . 9 |- ( ( ( A +P. 1P ) e. P. /\ B e. P. ) -> ( ( A +P. 1P ) .P. B ) = ( B .P. ( A +P. 1P ) ) )
19	5, 6, 18	syl2anc 411	. . . . . . . 8 |- ( ph -> ( ( A +P. 1P ) .P. B ) = ( B .P. ( A +P. 1P ) ) )
20		distrprg 7600	. . . . . . . . 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 1248	. . . . . . . 8 |- ( ph -> ( B .P. ( A +P. 1P ) ) = ( ( B .P. A ) +P. ( B .P. 1P ) ) )
22		mulcomprg 7592	. . . . . . . . . . 11 |- ( ( B e. P. /\ A e. P. ) -> ( B .P. A ) = ( A .P. B ) )
23	6, 1, 22	syl2anc 411	. . . . . . . . . 10 |- ( ph -> ( B .P. A ) = ( A .P. B ) )
24		recidpirqlemcalc.rec	. . . . . . . . . 10 |- ( ph -> ( A .P. B ) = 1P )
25	23, 24	eqtrd 2220	. . . . . . . . 9 |- ( ph -> ( B .P. A ) = 1P )
26		1idpr 7604	. . . . . . . . . 10 |- ( B e. P. -> ( B .P. 1P ) = B )
27	6, 26	syl 14	. . . . . . . . 9 |- ( ph -> ( B .P. 1P ) = B )
28	25, 27	oveq12d 5906	. . . . . . . 8 |- ( ph -> ( ( B .P. A ) +P. ( B .P. 1P ) ) = ( 1P +P. B ) )
29	19, 21, 28	3eqtrd 2224	. . . . . . 7 |- ( ph -> ( ( A +P. 1P ) .P. B ) = ( 1P +P. B ) )
30	29	oveq1d 5903	. . . . . 6 |- ( ph -> ( ( ( A +P. 1P ) .P. B ) +P. ( A +P. 1P ) ) = ( ( 1P +P. B ) +P. ( A +P. 1P ) ) )
31	14, 17, 30	3eqtrd 2224	. . . . 5 |- ( ph -> ( ( A +P. 1P ) .P. ( B +P. 1P ) ) = ( ( 1P +P. B ) +P. ( A +P. 1P ) ) )
32		1idpr 7604	. . . . . 6 |- ( 1P e. P. -> ( 1P .P. 1P ) = 1P )
33	2, 32	mp1i 10	. . . . 5 |- ( ph -> ( 1P .P. 1P ) = 1P )
34	31, 33	oveq12d 5906	. . . 4 |- ( ph -> ( ( ( A +P. 1P ) .P. ( B +P. 1P ) ) +P. ( 1P .P. 1P ) ) = ( ( ( 1P +P. B ) +P. ( A +P. 1P ) ) +P. 1P ) )
35		addcomprg 7590	. . . . . . . 8 |- ( ( 1P e. P. /\ B e. P. ) -> ( 1P +P. B ) = ( B +P. 1P ) )
36	3, 6, 35	syl2anc 411	. . . . . . 7 |- ( ph -> ( 1P +P. B ) = ( B +P. 1P ) )
37	36	oveq1d 5903	. . . . . 6 |- ( ph -> ( ( 1P +P. B ) +P. ( A +P. 1P ) ) = ( ( B +P. 1P ) +P. ( A +P. 1P ) ) )
38		addcomprg 7590	. . . . . . 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 411	. . . . . 6 |- ( ph -> ( ( B +P. 1P ) +P. ( A +P. 1P ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
40	37, 39	eqtrd 2220	. . . . 5 |- ( ph -> ( ( 1P +P. B ) +P. ( A +P. 1P ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
41	40	oveq1d 5903	. . . 4 |- ( ph -> ( ( ( 1P +P. B ) +P. ( A +P. 1P ) ) +P. 1P ) = ( ( ( A +P. 1P ) +P. ( B +P. 1P ) ) +P. 1P ) )
42	34, 41	eqtrd 2220	. . 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 5903	. 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 7592	. . . . . 6 |- ( ( 1P e. P. /\ ( B +P. 1P ) e. P. ) -> ( 1P .P. ( B +P. 1P ) ) = ( ( B +P. 1P ) .P. 1P ) )
45	3, 8, 44	syl2anc 411	. . . . 5 |- ( ph -> ( 1P .P. ( B +P. 1P ) ) = ( ( B +P. 1P ) .P. 1P ) )
46		1idpr 7604	. . . . . 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 2220	. . . 4 |- ( ph -> ( 1P .P. ( B +P. 1P ) ) = ( B +P. 1P ) )
49	16, 48	oveq12d 5906	. . 3 |- ( ph -> ( ( ( A +P. 1P ) .P. 1P ) +P. ( 1P .P. ( B +P. 1P ) ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
50	49	oveq1d 5903	. 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 2230	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 1363   e. wcel 2158  (class class class)co 5888   P.cnp 7303   1Pc1p 7304   +P. cpp 7305   .P. cmp 7306

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-2o 6431  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-enq0 7436  df-nq0 7437  df-0nq0 7438  df-plq0 7439  df-mq0 7440  df-inp 7478  df-i1p 7479  df-iplp 7480  df-imp 7481

This theorem is referenced by:  recidpirq  7870