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Theorem recidpirqlemcalc 7087
Description: Lemma for recidpirq 7088. 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 6806 . . . . . 6 1PP
32a1i 9 . . . . 5 (𝜑 → 1PP)
4 addclpr 6789 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
51, 3, 4syl2anc 403 . . . 4 (𝜑 → (𝐴 +P 1P) ∈ P)
6 recidpirqlemcalc.b . . . . 5 (𝜑𝐵P)
7 addclpr 6789 . . . . 5 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
86, 3, 7syl2anc 403 . . . 4 (𝜑 → (𝐵 +P 1P) ∈ P)
9 addclpr 6789 . . . 4 (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
105, 8, 9syl2anc 403 . . 3 (𝜑 → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
11 addassprg 6831 . . 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 1170 . 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 6840 . . . . . . 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 1170 . . . . . 6 (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)))
15 1idpr 6844 . . . . . . . 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 5559 . . . . . 6 (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)))
18 mulcomprg 6832 . . . . . . . . 9 (((𝐴 +P 1P) ∈ P𝐵P) → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
195, 6, 18syl2anc 403 . . . . . . . 8 (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
20 distrprg 6840 . . . . . . . . 9 ((𝐵P𝐴P ∧ 1PP) → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
216, 1, 3, 20syl3anc 1170 . . . . . . . 8 (𝜑 → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
22 mulcomprg 6832 . . . . . . . . . . 11 ((𝐵P𝐴P) → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
236, 1, 22syl2anc 403 . . . . . . . . . 10 (𝜑 → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
24 recidpirqlemcalc.rec . . . . . . . . . 10 (𝜑 → (𝐴 ·P 𝐵) = 1P)
2523, 24eqtrd 2114 . . . . . . . . 9 (𝜑 → (𝐵 ·P 𝐴) = 1P)
26 1idpr 6844 . . . . . . . . . 10 (𝐵P → (𝐵 ·P 1P) = 𝐵)
276, 26syl 14 . . . . . . . . 9 (𝜑 → (𝐵 ·P 1P) = 𝐵)
2825, 27oveq12d 5561 . . . . . . . 8 (𝜑 → ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)) = (1P +P 𝐵))
2919, 21, 283eqtrd 2118 . . . . . . 7 (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (1P +P 𝐵))
3029oveq1d 5558 . . . . . 6 (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
3114, 17, 303eqtrd 2118 . . . . 5 (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
32 1idpr 6844 . . . . . 6 (1PP → (1P ·P 1P) = 1P)
332, 32mp1i 10 . . . . 5 (𝜑 → (1P ·P 1P) = 1P)
3431, 33oveq12d 5561 . . . 4 (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P))
35 addcomprg 6830 . . . . . . . 8 ((1PP𝐵P) → (1P +P 𝐵) = (𝐵 +P 1P))
363, 6, 35syl2anc 403 . . . . . . 7 (𝜑 → (1P +P 𝐵) = (𝐵 +P 1P))
3736oveq1d 5558 . . . . . 6 (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐵 +P 1P) +P (𝐴 +P 1P)))
38 addcomprg 6830 . . . . . . 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 403 . . . . . 6 (𝜑 → ((𝐵 +P 1P) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
4037, 39eqtrd 2114 . . . . 5 (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
4140oveq1d 5558 . . . 4 (𝜑 → (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
4234, 41eqtrd 2114 . . 3 (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
4342oveq1d 5558 . 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 6832 . . . . . 6 ((1PP ∧ (𝐵 +P 1P) ∈ P) → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
453, 8, 44syl2anc 403 . . . . 5 (𝜑 → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
46 1idpr 6844 . . . . . 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 2114 . . . 4 (𝜑 → (1P ·P (𝐵 +P 1P)) = (𝐵 +P 1P))
4916, 48oveq12d 5561 . . 3 (𝜑 → (((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
5049oveq1d 5558 . 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 2124 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 1285  wcel 1434  (class class class)co 5543  Pcnp 6543  1Pc1p 6544   +P cpp 6545   ·P cmp 6546
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-i1p 6719  df-iplp 6720  df-imp 6721
This theorem is referenced by:  recidpirq  7088
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