Theorem recidpirqlemcalc 8067

Description: Lemma for recidpirq 8068. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.)

Hypotheses

Ref	Expression
recidpirqlemcalc.a	⊢ (𝜑 → 𝐴 ∈ P)
recidpirqlemcalc.b	⊢ (𝜑 → 𝐵 ∈ P)
recidpirqlemcalc.rec	⊢ (𝜑 → (𝐴 ·P 𝐵) = 1P)

Assertion

Ref	Expression
recidpirqlemcalc	⊢ (𝜑 → ((((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 ⊢ (𝜑 → 𝐴 ∈ P)
2		1pr 7764	. . . . . 6 ⊢ 1P ∈ P
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → 1P ∈ P)
4		addclpr 7747	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 +P 1P) ∈ P)
5	1, 3, 4	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐴 +P 1P) ∈ P)
6		recidpirqlemcalc.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ P)
7		addclpr 7747	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 1P ∈ P) → (𝐵 +P 1P) ∈ P)
8	6, 3, 7	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐵 +P 1P) ∈ P)
9		addclpr 7747	. . . 4 ⊢ (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
10	5, 8, 9	syl2anc 411	. . 3 ⊢ (𝜑 → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
11		addassprg 7789	. . 3 ⊢ ((((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P ∧ 1P ∈ P ∧ 1P ∈ 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 ⊢ (𝜑 → ((((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P (1P +P 1P)))
13		distrprg 7798	. . . . . . 7 ⊢ (((𝐴 +P 1P) ∈ P ∧ 𝐵 ∈ P ∧ 1P ∈ 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 ⊢ (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)))
15		1idpr 7802	. . . . . . . 8 ⊢ ((𝐴 +P 1P) ∈ P → ((𝐴 +P 1P) ·P 1P) = (𝐴 +P 1P))
16	5, 15	syl 14	. . . . . . 7 ⊢ (𝜑 → ((𝐴 +P 1P) ·P 1P) = (𝐴 +P 1P))
17	16	oveq2d 6029	. . . . . 6 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)))
18		mulcomprg 7790	. . . . . . . . 9 ⊢ (((𝐴 +P 1P) ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
19	5, 6, 18	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
20		distrprg 7798	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ 1P ∈ P) → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
21	6, 1, 3, 20	syl3anc 1271	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
22		mulcomprg 7790	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
23	6, 1, 22	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
24		recidpirqlemcalc.rec	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ·P 𝐵) = 1P)
25	23, 24	eqtrd 2262	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·P 𝐴) = 1P)
26		1idpr 7802	. . . . . . . . . 10 ⊢ (𝐵 ∈ P → (𝐵 ·P 1P) = 𝐵)
27	6, 26	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·P 1P) = 𝐵)
28	25, 27	oveq12d 6031	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)) = (1P +P 𝐵))
29	19, 21, 28	3eqtrd 2266	. . . . . . 7 ⊢ (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (1P +P 𝐵))
30	29	oveq1d 6028	. . . . . 6 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
31	14, 17, 30	3eqtrd 2266	. . . . 5 ⊢ (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
32		1idpr 7802	. . . . . 6 ⊢ (1P ∈ P → (1P ·P 1P) = 1P)
33	2, 32	mp1i 10	. . . . 5 ⊢ (𝜑 → (1P ·P 1P) = 1P)
34	31, 33	oveq12d 6031	. . . 4 ⊢ (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P))
35		addcomprg 7788	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 𝐵 ∈ P) → (1P +P 𝐵) = (𝐵 +P 1P))
36	3, 6, 35	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (1P +P 𝐵) = (𝐵 +P 1P))
37	36	oveq1d 6028	. . . . . 6 ⊢ (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐵 +P 1P) +P (𝐴 +P 1P)))
38		addcomprg 7788	. . . . . . 7 ⊢ (((𝐵 +P 1P) ∈ P ∧ (𝐴 +P 1P) ∈ P) → ((𝐵 +P 1P) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
39	8, 5, 38	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ((𝐵 +P 1P) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
40	37, 39	eqtrd 2262	. . . . 5 ⊢ (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
41	40	oveq1d 6028	. . . 4 ⊢ (𝜑 → (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
42	34, 41	eqtrd 2262	. . 3 ⊢ (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
43	42	oveq1d 6028	. 2 ⊢ (𝜑 → ((((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 7790	. . . . . 6 ⊢ ((1P ∈ P ∧ (𝐵 +P 1P) ∈ P) → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
45	3, 8, 44	syl2anc 411	. . . . 5 ⊢ (𝜑 → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
46		1idpr 7802	. . . . . 6 ⊢ ((𝐵 +P 1P) ∈ P → ((𝐵 +P 1P) ·P 1P) = (𝐵 +P 1P))
47	8, 46	syl 14	. . . . 5 ⊢ (𝜑 → ((𝐵 +P 1P) ·P 1P) = (𝐵 +P 1P))
48	45, 47	eqtrd 2262	. . . 4 ⊢ (𝜑 → (1P ·P (𝐵 +P 1P)) = (𝐵 +P 1P))
49	16, 48	oveq12d 6031	. . 3 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
50	49	oveq1d 6028	. 2 ⊢ (𝜑 → ((((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 ⊢ (𝜑 → ((((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   ∈ wcel 2200  (class class class)co 6013  Pcnp 7501  1_Pc1p 7502   +_P cpp 7503   ·_P cmp 7504

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-i1p 7677  df-iplp 7678  df-imp 7679

This theorem is referenced by:  recidpirq  8068