Theorem recidpirqlemcalc 7808

Description: Lemma for recidpirq 7809. 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 7505	. . . . . 6 |- 1P e. P.
3	2	a1i 9	. . . . 5 |- ( ph -> 1P e. P. )
4		addclpr 7488	. . . . 5 |- ( ( A e. P. /\ 1P e. P. ) -> ( A +P. 1P ) e. P. )
5	1, 3, 4	syl2anc 409	. . . 4 |- ( ph -> ( A +P. 1P ) e. P. )
6		recidpirqlemcalc.b	. . . . 5 |- ( ph -> B e. P. )
7		addclpr 7488	. . . . 5 |- ( ( B e. P. /\ 1P e. P. ) -> ( B +P. 1P ) e. P. )
8	6, 3, 7	syl2anc 409	. . . 4 |- ( ph -> ( B +P. 1P ) e. P. )
9		addclpr 7488	. . . 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 409	. . 3 |- ( ph -> ( ( A +P. 1P ) +P. ( B +P. 1P ) ) e. P. )
11		addassprg 7530	. . 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 1233	. 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 7539	. . . . . . 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 1233	. . . . . 6 |- ( ph -> ( ( A +P. 1P ) .P. ( B +P. 1P ) ) = ( ( ( A +P. 1P ) .P. B ) +P. ( ( A +P. 1P ) .P. 1P ) ) )
15		1idpr 7543	. . . . . . . 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 5867	. . . . . 6 |- ( ph -> ( ( ( A +P. 1P ) .P. B ) +P. ( ( A +P. 1P ) .P. 1P ) ) = ( ( ( A +P. 1P ) .P. B ) +P. ( A +P. 1P ) ) )
18		mulcomprg 7531	. . . . . . . . 9 |- ( ( ( A +P. 1P ) e. P. /\ B e. P. ) -> ( ( A +P. 1P ) .P. B ) = ( B .P. ( A +P. 1P ) ) )
19	5, 6, 18	syl2anc 409	. . . . . . . 8 |- ( ph -> ( ( A +P. 1P ) .P. B ) = ( B .P. ( A +P. 1P ) ) )
20		distrprg 7539	. . . . . . . . 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 1233	. . . . . . . 8 |- ( ph -> ( B .P. ( A +P. 1P ) ) = ( ( B .P. A ) +P. ( B .P. 1P ) ) )
22		mulcomprg 7531	. . . . . . . . . . 11 |- ( ( B e. P. /\ A e. P. ) -> ( B .P. A ) = ( A .P. B ) )
23	6, 1, 22	syl2anc 409	. . . . . . . . . 10 |- ( ph -> ( B .P. A ) = ( A .P. B ) )
24		recidpirqlemcalc.rec	. . . . . . . . . 10 |- ( ph -> ( A .P. B ) = 1P )
25	23, 24	eqtrd 2203	. . . . . . . . 9 |- ( ph -> ( B .P. A ) = 1P )
26		1idpr 7543	. . . . . . . . . 10 |- ( B e. P. -> ( B .P. 1P ) = B )
27	6, 26	syl 14	. . . . . . . . 9 |- ( ph -> ( B .P. 1P ) = B )
28	25, 27	oveq12d 5869	. . . . . . . 8 |- ( ph -> ( ( B .P. A ) +P. ( B .P. 1P ) ) = ( 1P +P. B ) )
29	19, 21, 28	3eqtrd 2207	. . . . . . 7 |- ( ph -> ( ( A +P. 1P ) .P. B ) = ( 1P +P. B ) )
30	29	oveq1d 5866	. . . . . 6 |- ( ph -> ( ( ( A +P. 1P ) .P. B ) +P. ( A +P. 1P ) ) = ( ( 1P +P. B ) +P. ( A +P. 1P ) ) )
31	14, 17, 30	3eqtrd 2207	. . . . 5 |- ( ph -> ( ( A +P. 1P ) .P. ( B +P. 1P ) ) = ( ( 1P +P. B ) +P. ( A +P. 1P ) ) )
32		1idpr 7543	. . . . . 6 |- ( 1P e. P. -> ( 1P .P. 1P ) = 1P )
33	2, 32	mp1i 10	. . . . 5 |- ( ph -> ( 1P .P. 1P ) = 1P )
34	31, 33	oveq12d 5869	. . . 4 |- ( ph -> ( ( ( A +P. 1P ) .P. ( B +P. 1P ) ) +P. ( 1P .P. 1P ) ) = ( ( ( 1P +P. B ) +P. ( A +P. 1P ) ) +P. 1P ) )
35		addcomprg 7529	. . . . . . . 8 |- ( ( 1P e. P. /\ B e. P. ) -> ( 1P +P. B ) = ( B +P. 1P ) )
36	3, 6, 35	syl2anc 409	. . . . . . 7 |- ( ph -> ( 1P +P. B ) = ( B +P. 1P ) )
37	36	oveq1d 5866	. . . . . 6 |- ( ph -> ( ( 1P +P. B ) +P. ( A +P. 1P ) ) = ( ( B +P. 1P ) +P. ( A +P. 1P ) ) )
38		addcomprg 7529	. . . . . . 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 409	. . . . . 6 |- ( ph -> ( ( B +P. 1P ) +P. ( A +P. 1P ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
40	37, 39	eqtrd 2203	. . . . 5 |- ( ph -> ( ( 1P +P. B ) +P. ( A +P. 1P ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
41	40	oveq1d 5866	. . . 4 |- ( ph -> ( ( ( 1P +P. B ) +P. ( A +P. 1P ) ) +P. 1P ) = ( ( ( A +P. 1P ) +P. ( B +P. 1P ) ) +P. 1P ) )
42	34, 41	eqtrd 2203	. . 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 5866	. 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 7531	. . . . . 6 |- ( ( 1P e. P. /\ ( B +P. 1P ) e. P. ) -> ( 1P .P. ( B +P. 1P ) ) = ( ( B +P. 1P ) .P. 1P ) )
45	3, 8, 44	syl2anc 409	. . . . 5 |- ( ph -> ( 1P .P. ( B +P. 1P ) ) = ( ( B +P. 1P ) .P. 1P ) )
46		1idpr 7543	. . . . . 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 2203	. . . 4 |- ( ph -> ( 1P .P. ( B +P. 1P ) ) = ( B +P. 1P ) )
49	16, 48	oveq12d 5869	. . 3 |- ( ph -> ( ( ( A +P. 1P ) .P. 1P ) +P. ( 1P .P. ( B +P. 1P ) ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
50	49	oveq1d 5866	. 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 2213	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 1348   e. wcel 2141  (class class class)co 5851   P.cnp 7242   1Pc1p 7243   +P. cpp 7244   .P. cmp 7245

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-1o 6393  df-2o 6394  df-oadd 6397  df-omul 6398  df-er 6510  df-ec 6512  df-qs 6516  df-ni 7255  df-pli 7256  df-mi 7257  df-lti 7258  df-plpq 7295  df-mpq 7296  df-enq 7298  df-nqqs 7299  df-plqqs 7300  df-mqqs 7301  df-1nqqs 7302  df-rq 7303  df-ltnqqs 7304  df-enq0 7375  df-nq0 7376  df-0nq0 7377  df-plq0 7378  df-mq0 7379  df-inp 7417  df-i1p 7418  df-iplp 7419  df-imp 7420

This theorem is referenced by:  recidpirq  7809