Theorem recidpirqlemcalc 7025

Description: Lemma for recidpirq 7026. 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 6744	. . . . . 6 ⊢ 1P ∈ P
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → 1P ∈ P)
4		addclpr 6727	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 +P 1P) ∈ P)
5	1, 3, 4	syl2anc 403	. . . 4 ⊢ (𝜑 → (𝐴 +P 1P) ∈ P)
6		recidpirqlemcalc.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ P)
7		addclpr 6727	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 1P ∈ P) → (𝐵 +P 1P) ∈ P)
8	6, 3, 7	syl2anc 403	. . . 4 ⊢ (𝜑 → (𝐵 +P 1P) ∈ P)
9		addclpr 6727	. . . 4 ⊢ (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
10	5, 8, 9	syl2anc 403	. . 3 ⊢ (𝜑 → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
11		addassprg 6769	. . 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 1169	. 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 6778	. . . . . . 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 1169	. . . . . 6 ⊢ (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)))
15		1idpr 6782	. . . . . . . 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 5548	. . . . . 6 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)))
18		mulcomprg 6770	. . . . . . . . 9 ⊢ (((𝐴 +P 1P) ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
19	5, 6, 18	syl2anc 403	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
20		distrprg 6778	. . . . . . . . 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 1169	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
22		mulcomprg 6770	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
23	6, 1, 22	syl2anc 403	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
24		recidpirqlemcalc.rec	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ·P 𝐵) = 1P)
25	23, 24	eqtrd 2113	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·P 𝐴) = 1P)
26		1idpr 6782	. . . . . . . . . 10 ⊢ (𝐵 ∈ P → (𝐵 ·P 1P) = 𝐵)
27	6, 26	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·P 1P) = 𝐵)
28	25, 27	oveq12d 5550	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)) = (1P +P 𝐵))
29	19, 21, 28	3eqtrd 2117	. . . . . . 7 ⊢ (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (1P +P 𝐵))
30	29	oveq1d 5547	. . . . . 6 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
31	14, 17, 30	3eqtrd 2117	. . . . 5 ⊢ (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
32		1idpr 6782	. . . . . 6 ⊢ (1P ∈ P → (1P ·P 1P) = 1P)
33	2, 32	mp1i 10	. . . . 5 ⊢ (𝜑 → (1P ·P 1P) = 1P)
34	31, 33	oveq12d 5550	. . . 4 ⊢ (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P))
35		addcomprg 6768	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 𝐵 ∈ P) → (1P +P 𝐵) = (𝐵 +P 1P))
36	3, 6, 35	syl2anc 403	. . . . . . 7 ⊢ (𝜑 → (1P +P 𝐵) = (𝐵 +P 1P))
37	36	oveq1d 5547	. . . . . 6 ⊢ (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐵 +P 1P) +P (𝐴 +P 1P)))
38		addcomprg 6768	. . . . . . 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 403	. . . . . 6 ⊢ (𝜑 → ((𝐵 +P 1P) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
40	37, 39	eqtrd 2113	. . . . 5 ⊢ (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
41	40	oveq1d 5547	. . . 4 ⊢ (𝜑 → (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
42	34, 41	eqtrd 2113	. . 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 5547	. 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 6770	. . . . . 6 ⊢ ((1P ∈ P ∧ (𝐵 +P 1P) ∈ P) → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
45	3, 8, 44	syl2anc 403	. . . . 5 ⊢ (𝜑 → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
46		1idpr 6782	. . . . . 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 2113	. . . 4 ⊢ (𝜑 → (1P ·P (𝐵 +P 1P)) = (𝐵 +P 1P))
49	16, 48	oveq12d 5550	. . 3 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
50	49	oveq1d 5547	. 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 2123	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 1284   ∈ wcel 1433  (class class class)co 5532  Pcnp 6481  1_Pc1p 6482   +_P cpp 6483   ·_P cmp 6484

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-i1p 6657  df-iplp 6658  df-imp 6659

This theorem is referenced by:  recidpirq  7026