Theorem recidpirqlemcalc 8040

Description: Lemma for recidpirq 8041. 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 7737	. . . . . 6 |- 1P e. P.
3	2	a1i 9	. . . . 5 |- ( ph -> 1P e. P. )
4		addclpr 7720	. . . . 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 7720	. . . . 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 7720	. . . 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 7762	. . 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 1271	. 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 7771	. . . . . . 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 1271	. . . . . 6 |- ( ph -> ( ( A +P. 1P ) .P. ( B +P. 1P ) ) = ( ( ( A +P. 1P ) .P. B ) +P. ( ( A +P. 1P ) .P. 1P ) ) )
15		1idpr 7775	. . . . . . . 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 6016	. . . . . 6 |- ( ph -> ( ( ( A +P. 1P ) .P. B ) +P. ( ( A +P. 1P ) .P. 1P ) ) = ( ( ( A +P. 1P ) .P. B ) +P. ( A +P. 1P ) ) )
18		mulcomprg 7763	. . . . . . . . 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 7771	. . . . . . . . 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 1271	. . . . . . . 8 |- ( ph -> ( B .P. ( A +P. 1P ) ) = ( ( B .P. A ) +P. ( B .P. 1P ) ) )
22		mulcomprg 7763	. . . . . . . . . . 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 2262	. . . . . . . . 9 |- ( ph -> ( B .P. A ) = 1P )
26		1idpr 7775	. . . . . . . . . 10 |- ( B e. P. -> ( B .P. 1P ) = B )
27	6, 26	syl 14	. . . . . . . . 9 |- ( ph -> ( B .P. 1P ) = B )
28	25, 27	oveq12d 6018	. . . . . . . 8 |- ( ph -> ( ( B .P. A ) +P. ( B .P. 1P ) ) = ( 1P +P. B ) )
29	19, 21, 28	3eqtrd 2266	. . . . . . 7 |- ( ph -> ( ( A +P. 1P ) .P. B ) = ( 1P +P. B ) )
30	29	oveq1d 6015	. . . . . 6 |- ( ph -> ( ( ( A +P. 1P ) .P. B ) +P. ( A +P. 1P ) ) = ( ( 1P +P. B ) +P. ( A +P. 1P ) ) )
31	14, 17, 30	3eqtrd 2266	. . . . 5 |- ( ph -> ( ( A +P. 1P ) .P. ( B +P. 1P ) ) = ( ( 1P +P. B ) +P. ( A +P. 1P ) ) )
32		1idpr 7775	. . . . . 6 |- ( 1P e. P. -> ( 1P .P. 1P ) = 1P )
33	2, 32	mp1i 10	. . . . 5 |- ( ph -> ( 1P .P. 1P ) = 1P )
34	31, 33	oveq12d 6018	. . . 4 |- ( ph -> ( ( ( A +P. 1P ) .P. ( B +P. 1P ) ) +P. ( 1P .P. 1P ) ) = ( ( ( 1P +P. B ) +P. ( A +P. 1P ) ) +P. 1P ) )
35		addcomprg 7761	. . . . . . . 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 6015	. . . . . 6 |- ( ph -> ( ( 1P +P. B ) +P. ( A +P. 1P ) ) = ( ( B +P. 1P ) +P. ( A +P. 1P ) ) )
38		addcomprg 7761	. . . . . . 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 2262	. . . . 5 |- ( ph -> ( ( 1P +P. B ) +P. ( A +P. 1P ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
41	40	oveq1d 6015	. . . 4 |- ( ph -> ( ( ( 1P +P. B ) +P. ( A +P. 1P ) ) +P. 1P ) = ( ( ( A +P. 1P ) +P. ( B +P. 1P ) ) +P. 1P ) )
42	34, 41	eqtrd 2262	. . 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 6015	. 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 7763	. . . . . 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 7775	. . . . . 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 2262	. . . 4 |- ( ph -> ( 1P .P. ( B +P. 1P ) ) = ( B +P. 1P ) )
49	16, 48	oveq12d 6018	. . 3 |- ( ph -> ( ( ( A +P. 1P ) .P. 1P ) +P. ( 1P .P. ( B +P. 1P ) ) ) = ( ( A +P. 1P ) +P. ( B +P. 1P ) ) )
50	49	oveq1d 6015	. 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 2272	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 1395   e. wcel 2200  (class class class)co 6000   P.cnp 7474   1Pc1p 7475   +P. cpp 7476   .P. cmp 7477

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-2o 6561  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-enq0 7607  df-nq0 7608  df-0nq0 7609  df-plq0 7610  df-mq0 7611  df-inp 7649  df-i1p 7650  df-iplp 7651  df-imp 7652

This theorem is referenced by:  recidpirq  8041