Theorem recidpirqlemcalc 8214

Description: Lemma for recidpirq 8215. 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 7911	. . . . . 6 ⊢ 1P ∈ P
3	2	a1i 9	. . . . 5 ⊢ (𝜑 → 1P ∈ P)
4		addclpr 7894	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 +P 1P) ∈ P)
5	1, 3, 4	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝐴 +P 1P) ∈ P)
6		recidpirqlemcalc.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ P)
7		addclpr 7894	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 1P ∈ P) → (𝐵 +P 1P) ∈ P)
8	6, 3, 7	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝐵 +P 1P) ∈ P)
9		addclpr 7894	. . . 4 ⊢ (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
10	5, 8, 9	syl2anc 415	. . 3 ⊢ (𝜑 → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
11		addassprg 7936	. . 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 1278	. 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 7945	. . . . . . 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 1278	. . . . . 6 ⊢ (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)))
15		1idpr 7949	. . . . . . . 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 6091	. . . . . 6 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)))
18		mulcomprg 7937	. . . . . . . . 9 ⊢ (((𝐴 +P 1P) ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
19	5, 6, 18	syl2anc 415	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
20		distrprg 7945	. . . . . . . . 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 1278	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
22		mulcomprg 7937	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
23	6, 1, 22	syl2anc 415	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
24		recidpirqlemcalc.rec	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ·P 𝐵) = 1P)
25	23, 24	eqtrd 2271	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·P 𝐴) = 1P)
26		1idpr 7949	. . . . . . . . . 10 ⊢ (𝐵 ∈ P → (𝐵 ·P 1P) = 𝐵)
27	6, 26	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·P 1P) = 𝐵)
28	25, 27	oveq12d 6093	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)) = (1P +P 𝐵))
29	19, 21, 28	3eqtrd 2275	. . . . . . 7 ⊢ (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (1P +P 𝐵))
30	29	oveq1d 6090	. . . . . 6 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
31	14, 17, 30	3eqtrd 2275	. . . . 5 ⊢ (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
32		1idpr 7949	. . . . . 6 ⊢ (1P ∈ P → (1P ·P 1P) = 1P)
33	2, 32	mp1i 10	. . . . 5 ⊢ (𝜑 → (1P ·P 1P) = 1P)
34	31, 33	oveq12d 6093	. . . 4 ⊢ (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P))
35		addcomprg 7935	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 𝐵 ∈ P) → (1P +P 𝐵) = (𝐵 +P 1P))
36	3, 6, 35	syl2anc 415	. . . . . . 7 ⊢ (𝜑 → (1P +P 𝐵) = (𝐵 +P 1P))
37	36	oveq1d 6090	. . . . . 6 ⊢ (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐵 +P 1P) +P (𝐴 +P 1P)))
38		addcomprg 7935	. . . . . . 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 415	. . . . . 6 ⊢ (𝜑 → ((𝐵 +P 1P) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
40	37, 39	eqtrd 2271	. . . . 5 ⊢ (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
41	40	oveq1d 6090	. . . 4 ⊢ (𝜑 → (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
42	34, 41	eqtrd 2271	. . 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 6090	. 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 7937	. . . . . 6 ⊢ ((1P ∈ P ∧ (𝐵 +P 1P) ∈ P) → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
45	3, 8, 44	syl2anc 415	. . . . 5 ⊢ (𝜑 → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
46		1idpr 7949	. . . . . 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 2271	. . . 4 ⊢ (𝜑 → (1P ·P (𝐵 +P 1P)) = (𝐵 +P 1P))
49	16, 48	oveq12d 6093	. . 3 ⊢ (𝜑 → (((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
50	49	oveq1d 6090	. 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 2281	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 1402   ∈ wcel 2209  (class class class)co 6075  Pcnp 7648  1_Pc1p 7649   +_P cpp 7650   ·_P cmp 7651

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-i1p 7824  df-iplp 7825  df-imp 7826

This theorem is referenced by:  recidpirq  8215