Theorem recidpirqlemcalc 7985

Description: Lemma for recidpirq 7986. 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 7682	. . . . . 6 ⊢ 1P ∈ P
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → 1P ∈ P)
4		addclpr 7665	. . . . 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 7665	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 1P ∈ P) → (𝐵 +P 1P) ∈ P)
8	6, 3, 7	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐵 +P 1P) ∈ P)
9		addclpr 7665	. . . 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 7707	. . 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 1250	. 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 7716	. . . . . . 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 1250	. . . . . 6 ⊢ (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)))
15		1idpr 7720	. . . . . . . 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 5972	. . . . . 6 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)))
18		mulcomprg 7708	. . . . . . . . 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 7716	. . . . . . . . 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 1250	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
22		mulcomprg 7708	. . . . . . . . . . 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 2239	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·P 𝐴) = 1P)
26		1idpr 7720	. . . . . . . . . 10 ⊢ (𝐵 ∈ P → (𝐵 ·P 1P) = 𝐵)
27	6, 26	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·P 1P) = 𝐵)
28	25, 27	oveq12d 5974	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)) = (1P +P 𝐵))
29	19, 21, 28	3eqtrd 2243	. . . . . . 7 ⊢ (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (1P +P 𝐵))
30	29	oveq1d 5971	. . . . . 6 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
31	14, 17, 30	3eqtrd 2243	. . . . 5 ⊢ (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
32		1idpr 7720	. . . . . 6 ⊢ (1P ∈ P → (1P ·P 1P) = 1P)
33	2, 32	mp1i 10	. . . . 5 ⊢ (𝜑 → (1P ·P 1P) = 1P)
34	31, 33	oveq12d 5974	. . . 4 ⊢ (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P))
35		addcomprg 7706	. . . . . . . 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 5971	. . . . . 6 ⊢ (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐵 +P 1P) +P (𝐴 +P 1P)))
38		addcomprg 7706	. . . . . . 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 2239	. . . . 5 ⊢ (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
41	40	oveq1d 5971	. . . 4 ⊢ (𝜑 → (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
42	34, 41	eqtrd 2239	. . 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 5971	. 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 7708	. . . . . 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 7720	. . . . . 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 2239	. . . 4 ⊢ (𝜑 → (1P ·P (𝐵 +P 1P)) = (𝐵 +P 1P))
49	16, 48	oveq12d 5974	. . 3 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
50	49	oveq1d 5971	. 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 2249	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 1373   ∈ wcel 2177  (class class class)co 5956  Pcnp 7419  1_Pc1p 7420   +_P cpp 7421   ·_P cmp 7422

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-eprel 4343  df-id 4347  df-po 4350  df-iso 4351  df-iord 4420  df-on 4422  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-irdg 6468  df-1o 6514  df-2o 6515  df-oadd 6518  df-omul 6519  df-er 6632  df-ec 6634  df-qs 6638  df-ni 7432  df-pli 7433  df-mi 7434  df-lti 7435  df-plpq 7472  df-mpq 7473  df-enq 7475  df-nqqs 7476  df-plqqs 7477  df-mqqs 7478  df-1nqqs 7479  df-rq 7480  df-ltnqqs 7481  df-enq0 7552  df-nq0 7553  df-0nq0 7554  df-plq0 7555  df-mq0 7556  df-inp 7594  df-i1p 7595  df-iplp 7596  df-imp 7597

This theorem is referenced by:  recidpirq  7986