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Theorem recidpirqlemcalc 7633
Description: Lemma for recidpirq 7634. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.)
Hypotheses
Ref Expression
recidpirqlemcalc.a (𝜑𝐴P)
recidpirqlemcalc.b (𝜑𝐵P)
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
StepHypRef Expression
1 recidpirqlemcalc.a . . . . 5 (𝜑𝐴P)
2 1pr 7330 . . . . . 6 1PP
32a1i 9 . . . . 5 (𝜑 → 1PP)
4 addclpr 7313 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
51, 3, 4syl2anc 408 . . . 4 (𝜑 → (𝐴 +P 1P) ∈ P)
6 recidpirqlemcalc.b . . . . 5 (𝜑𝐵P)
7 addclpr 7313 . . . . 5 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
86, 3, 7syl2anc 408 . . . 4 (𝜑 → (𝐵 +P 1P) ∈ P)
9 addclpr 7313 . . . 4 (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
105, 8, 9syl2anc 408 . . 3 (𝜑 → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
11 addassprg 7355 . . 3 ((((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P ∧ 1PP ∧ 1PP) → ((((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P (1P +P 1P)))
1210, 3, 3, 11syl3anc 1201 . 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 7364 . . . . . . 7 (((𝐴 +P 1P) ∈ P𝐵P ∧ 1PP) → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)))
145, 6, 3, 13syl3anc 1201 . . . . . 6 (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)))
15 1idpr 7368 . . . . . . . 8 ((𝐴 +P 1P) ∈ P → ((𝐴 +P 1P) ·P 1P) = (𝐴 +P 1P))
165, 15syl 14 . . . . . . 7 (𝜑 → ((𝐴 +P 1P) ·P 1P) = (𝐴 +P 1P))
1716oveq2d 5758 . . . . . 6 (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)))
18 mulcomprg 7356 . . . . . . . . 9 (((𝐴 +P 1P) ∈ P𝐵P) → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
195, 6, 18syl2anc 408 . . . . . . . 8 (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
20 distrprg 7364 . . . . . . . . 9 ((𝐵P𝐴P ∧ 1PP) → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
216, 1, 3, 20syl3anc 1201 . . . . . . . 8 (𝜑 → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
22 mulcomprg 7356 . . . . . . . . . . 11 ((𝐵P𝐴P) → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
236, 1, 22syl2anc 408 . . . . . . . . . 10 (𝜑 → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
24 recidpirqlemcalc.rec . . . . . . . . . 10 (𝜑 → (𝐴 ·P 𝐵) = 1P)
2523, 24eqtrd 2150 . . . . . . . . 9 (𝜑 → (𝐵 ·P 𝐴) = 1P)
26 1idpr 7368 . . . . . . . . . 10 (𝐵P → (𝐵 ·P 1P) = 𝐵)
276, 26syl 14 . . . . . . . . 9 (𝜑 → (𝐵 ·P 1P) = 𝐵)
2825, 27oveq12d 5760 . . . . . . . 8 (𝜑 → ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)) = (1P +P 𝐵))
2919, 21, 283eqtrd 2154 . . . . . . 7 (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (1P +P 𝐵))
3029oveq1d 5757 . . . . . 6 (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
3114, 17, 303eqtrd 2154 . . . . 5 (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
32 1idpr 7368 . . . . . 6 (1PP → (1P ·P 1P) = 1P)
332, 32mp1i 10 . . . . 5 (𝜑 → (1P ·P 1P) = 1P)
3431, 33oveq12d 5760 . . . 4 (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P))
35 addcomprg 7354 . . . . . . . 8 ((1PP𝐵P) → (1P +P 𝐵) = (𝐵 +P 1P))
363, 6, 35syl2anc 408 . . . . . . 7 (𝜑 → (1P +P 𝐵) = (𝐵 +P 1P))
3736oveq1d 5757 . . . . . 6 (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐵 +P 1P) +P (𝐴 +P 1P)))
38 addcomprg 7354 . . . . . . 7 (((𝐵 +P 1P) ∈ P ∧ (𝐴 +P 1P) ∈ P) → ((𝐵 +P 1P) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
398, 5, 38syl2anc 408 . . . . . 6 (𝜑 → ((𝐵 +P 1P) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
4037, 39eqtrd 2150 . . . . 5 (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
4140oveq1d 5757 . . . 4 (𝜑 → (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
4234, 41eqtrd 2150 . . 3 (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
4342oveq1d 5757 . 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 7356 . . . . . 6 ((1PP ∧ (𝐵 +P 1P) ∈ P) → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
453, 8, 44syl2anc 408 . . . . 5 (𝜑 → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
46 1idpr 7368 . . . . . 6 ((𝐵 +P 1P) ∈ P → ((𝐵 +P 1P) ·P 1P) = (𝐵 +P 1P))
478, 46syl 14 . . . . 5 (𝜑 → ((𝐵 +P 1P) ·P 1P) = (𝐵 +P 1P))
4845, 47eqtrd 2150 . . . 4 (𝜑 → (1P ·P (𝐵 +P 1P)) = (𝐵 +P 1P))
4916, 48oveq12d 5760 . . 3 (𝜑 → (((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
5049oveq1d 5757 . 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)))
5112, 43, 503eqtr4d 2160 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)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  (class class class)co 5742  Pcnp 7067  1Pc1p 7068   +P cpp 7069   ·P cmp 7070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-i1p 7243  df-iplp 7244  df-imp 7245
This theorem is referenced by:  recidpirq  7634
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