Theorem recidpirqlemcalc 7941

Description: Lemma for recidpirq 7942. 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 7638	. . . . . 6 ⊢ 1P ∈ P
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → 1P ∈ P)
4		addclpr 7621	. . . . 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 7621	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 1P ∈ P) → (𝐵 +P 1P) ∈ P)
8	6, 3, 7	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐵 +P 1P) ∈ P)
9		addclpr 7621	. . . 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 7663	. . 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 1249	. 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 7672	. . . . . . 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 1249	. . . . . 6 ⊢ (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)))
15		1idpr 7676	. . . . . . . 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 5941	. . . . . 6 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)))
18		mulcomprg 7664	. . . . . . . . 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 7672	. . . . . . . . 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 1249	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
22		mulcomprg 7664	. . . . . . . . . . 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 2229	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·P 𝐴) = 1P)
26		1idpr 7676	. . . . . . . . . 10 ⊢ (𝐵 ∈ P → (𝐵 ·P 1P) = 𝐵)
27	6, 26	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·P 1P) = 𝐵)
28	25, 27	oveq12d 5943	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)) = (1P +P 𝐵))
29	19, 21, 28	3eqtrd 2233	. . . . . . 7 ⊢ (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (1P +P 𝐵))
30	29	oveq1d 5940	. . . . . 6 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
31	14, 17, 30	3eqtrd 2233	. . . . 5 ⊢ (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
32		1idpr 7676	. . . . . 6 ⊢ (1P ∈ P → (1P ·P 1P) = 1P)
33	2, 32	mp1i 10	. . . . 5 ⊢ (𝜑 → (1P ·P 1P) = 1P)
34	31, 33	oveq12d 5943	. . . 4 ⊢ (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P))
35		addcomprg 7662	. . . . . . . 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 5940	. . . . . 6 ⊢ (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐵 +P 1P) +P (𝐴 +P 1P)))
38		addcomprg 7662	. . . . . . 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 2229	. . . . 5 ⊢ (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
41	40	oveq1d 5940	. . . 4 ⊢ (𝜑 → (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
42	34, 41	eqtrd 2229	. . 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 5940	. 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 7664	. . . . . 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 7676	. . . . . 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 2229	. . . 4 ⊢ (𝜑 → (1P ·P (𝐵 +P 1P)) = (𝐵 +P 1P))
49	16, 48	oveq12d 5943	. . 3 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
50	49	oveq1d 5940	. 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 2239	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 1364   ∈ wcel 2167  (class class class)co 5925  Pcnp 7375  1_Pc1p 7376   +_P cpp 7377   ·_P cmp 7378

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-i1p 7551  df-iplp 7552  df-imp 7553

This theorem is referenced by:  recidpirq  7942