Theorem recidpirqlemcalc 7924

Description: Lemma for recidpirq 7925. 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 7621	. . . . . 6 |- 1P e. P.
3	2	a1i 9	. . . . 5 |- ( ph -> 1P e. P. )
4		addclpr 7604	. . . . 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 7604	. . . . 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 7604	. . . 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 7646	. . 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 1249	. 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 7655	. . . . . . 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 1249	. . . . . 6 |- ( ph -> ( ( A +P. 1P ) .P. ( B +P. 1P ) ) = ( ( ( A +P. 1P ) .P. B ) +P. ( ( A +P. 1P ) .P. 1P ) ) )
15		1idpr 7659	. . . . . . . 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 5938	. . . . . 6 |- ( ph -> ( ( ( A +P. 1P ) .P. B ) +P. ( ( A +P. 1P ) .P. 1P ) ) = ( ( ( A +P. 1P ) .P. B ) +P. ( A +P. 1P ) ) )
18		mulcomprg 7647	. . . . . . . . 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 7655	. . . . . . . . 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 1249	. . . . . . . 8 |- ( ph -> ( B .P. ( A +P. 1P ) ) = ( ( B .P. A ) +P. ( B .P. 1P ) ) )
22		mulcomprg 7647	. . . . . . . . . . 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 2229	. . . . . . . . 9 |- ( ph -> ( B .P. A ) = 1P )
26		1idpr 7659	. . . . . . . . . 10 |- ( B e. P. -> ( B .P. 1P ) = B )
27	6, 26	syl 14	. . . . . . . . 9 |- ( ph -> ( B .P. 1P ) = B )
28	25, 27	oveq12d 5940	. . . . . . . 8 |- ( ph -> ( ( B .P. A ) +P. ( B .P. 1P ) ) = ( 1P +P. B ) )
29	19, 21, 28	3eqtrd 2233	. . . . . . 7 |- ( ph -> ( ( A +P. 1P ) .P. B ) = ( 1P +P. B ) )
30	29	oveq1d 5937	. . . . . 6 |- ( ph -> ( ( ( A +P. 1P ) .P. B ) +P. ( A +P. 1P ) ) = ( ( 1P +P. B ) +P. ( A +P. 1P ) ) )
31	14, 17, 30	3eqtrd 2233	. . . . 5 |- ( ph -> ( ( A +P. 1P ) .P. ( B +P. 1P ) ) = ( ( 1P +P. B ) +P. ( A +P. 1P ) ) )
32		1idpr 7659	. . . . . 6 |- ( 1P e. P. -> ( 1P .P. 1P ) = 1P )
33	2, 32	mp1i 10	. . . . 5 |- ( ph -> ( 1P .P. 1P ) = 1P )
34	31, 33	oveq12d 5940	. . . 4 |- ( ph -> ( ( ( A +P. 1P ) .P. ( B +P. 1P ) ) +P. ( 1P .P. 1P ) ) = ( ( ( 1P +P. B ) +P. ( A +P. 1P ) ) +P. 1P ) )
35		addcomprg 7645	. . . . . . . 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 5937	. . . . . 6 |- ( ph -> ( ( 1P +P. B ) +P. ( A +P. 1P ) ) = ( ( B +P. 1P ) +P. ( A +P. 1P ) ) )
38		addcomprg 7645	. . . . . . 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 2229	. . . . 5 |- ( ph -> ( ( 1P +P. B ) +P. ( A +P. 1P ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
41	40	oveq1d 5937	. . . 4 |- ( ph -> ( ( ( 1P +P. B ) +P. ( A +P. 1P ) ) +P. 1P ) = ( ( ( A +P. 1P ) +P. ( B +P. 1P ) ) +P. 1P ) )
42	34, 41	eqtrd 2229	. . 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 5937	. 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 7647	. . . . . 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 7659	. . . . . 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 2229	. . . 4 |- ( ph -> ( 1P .P. ( B +P. 1P ) ) = ( B +P. 1P ) )
49	16, 48	oveq12d 5940	. . 3 |- ( ph -> ( ( ( A +P. 1P ) .P. 1P ) +P. ( 1P .P. ( B +P. 1P ) ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
50	49	oveq1d 5937	. 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 2239	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 1364   e. wcel 2167  (class class class)co 5922   P.cnp 7358   1Pc1p 7359   +P. cpp 7360   .P. cmp 7361

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-i1p 7534  df-iplp 7535  df-imp 7536

This theorem is referenced by:  recidpirq  7925