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Theorem recidpirqlemcalc 6931
Description: Lemma for recidpirq 6932. 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 6650 . . . . . 6 1PP
32a1i 9 . . . . 5 (𝜑 → 1PP)
4 addclpr 6633 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
51, 3, 4syl2anc 391 . . . 4 (𝜑 → (𝐴 +P 1P) ∈ P)
6 recidpirqlemcalc.b . . . . 5 (𝜑𝐵P)
7 addclpr 6633 . . . . 5 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
86, 3, 7syl2anc 391 . . . 4 (𝜑 → (𝐵 +P 1P) ∈ P)
9 addclpr 6633 . . . 4 (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
105, 8, 9syl2anc 391 . . 3 (𝜑 → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
11 addassprg 6675 . . 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 1135 . 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 6684 . . . . . . 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 1135 . . . . . 6 (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)))
15 1idpr 6688 . . . . . . . 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 5528 . . . . . 6 (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)))
18 mulcomprg 6676 . . . . . . . . 9 (((𝐴 +P 1P) ∈ P𝐵P) → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
195, 6, 18syl2anc 391 . . . . . . . 8 (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
20 distrprg 6684 . . . . . . . . 9 ((𝐵P𝐴P ∧ 1PP) → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
216, 1, 3, 20syl3anc 1135 . . . . . . . 8 (𝜑 → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
22 mulcomprg 6676 . . . . . . . . . . 11 ((𝐵P𝐴P) → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
236, 1, 22syl2anc 391 . . . . . . . . . 10 (𝜑 → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
24 recidpirqlemcalc.rec . . . . . . . . . 10 (𝜑 → (𝐴 ·P 𝐵) = 1P)
2523, 24eqtrd 2072 . . . . . . . . 9 (𝜑 → (𝐵 ·P 𝐴) = 1P)
26 1idpr 6688 . . . . . . . . . 10 (𝐵P → (𝐵 ·P 1P) = 𝐵)
276, 26syl 14 . . . . . . . . 9 (𝜑 → (𝐵 ·P 1P) = 𝐵)
2825, 27oveq12d 5530 . . . . . . . 8 (𝜑 → ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)) = (1P +P 𝐵))
2919, 21, 283eqtrd 2076 . . . . . . 7 (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (1P +P 𝐵))
3029oveq1d 5527 . . . . . 6 (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
3114, 17, 303eqtrd 2076 . . . . 5 (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
32 1idpr 6688 . . . . . 6 (1PP → (1P ·P 1P) = 1P)
332, 32mp1i 10 . . . . 5 (𝜑 → (1P ·P 1P) = 1P)
3431, 33oveq12d 5530 . . . 4 (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P))
35 addcomprg 6674 . . . . . . . 8 ((1PP𝐵P) → (1P +P 𝐵) = (𝐵 +P 1P))
363, 6, 35syl2anc 391 . . . . . . 7 (𝜑 → (1P +P 𝐵) = (𝐵 +P 1P))
3736oveq1d 5527 . . . . . 6 (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐵 +P 1P) +P (𝐴 +P 1P)))
38 addcomprg 6674 . . . . . . 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 391 . . . . . 6 (𝜑 → ((𝐵 +P 1P) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
4037, 39eqtrd 2072 . . . . 5 (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
4140oveq1d 5527 . . . 4 (𝜑 → (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
4234, 41eqtrd 2072 . . 3 (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
4342oveq1d 5527 . 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 6676 . . . . . 6 ((1PP ∧ (𝐵 +P 1P) ∈ P) → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
453, 8, 44syl2anc 391 . . . . 5 (𝜑 → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
46 1idpr 6688 . . . . . 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 2072 . . . 4 (𝜑 → (1P ·P (𝐵 +P 1P)) = (𝐵 +P 1P))
4916, 48oveq12d 5530 . . 3 (𝜑 → (((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
5049oveq1d 5527 . 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 2082 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 1243  wcel 1393  (class class class)co 5512  Pcnp 6387  1Pc1p 6388   +P cpp 6389   ·P cmp 6390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-imp 6565
This theorem is referenced by:  recidpirq  6932
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