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Theorem recidpirqlemcalc 6961
Description: Lemma for recidpirq 6962. 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 6680 . . . . . 6 1PP
32a1i 9 . . . . 5 (𝜑 → 1PP)
4 addclpr 6663 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
51, 3, 4syl2anc 397 . . . 4 (𝜑 → (𝐴 +P 1P) ∈ P)
6 recidpirqlemcalc.b . . . . 5 (𝜑𝐵P)
7 addclpr 6663 . . . . 5 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
86, 3, 7syl2anc 397 . . . 4 (𝜑 → (𝐵 +P 1P) ∈ P)
9 addclpr 6663 . . . 4 (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
105, 8, 9syl2anc 397 . . 3 (𝜑 → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
11 addassprg 6705 . . 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 1144 . 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 6714 . . . . . . 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 1144 . . . . . 6 (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)))
15 1idpr 6718 . . . . . . . 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 5553 . . . . . 6 (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)))
18 mulcomprg 6706 . . . . . . . . 9 (((𝐴 +P 1P) ∈ P𝐵P) → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
195, 6, 18syl2anc 397 . . . . . . . 8 (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (𝐵 ·P (𝐴 +P 1P)))
20 distrprg 6714 . . . . . . . . 9 ((𝐵P𝐴P ∧ 1PP) → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
216, 1, 3, 20syl3anc 1144 . . . . . . . 8 (𝜑 → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
22 mulcomprg 6706 . . . . . . . . . . 11 ((𝐵P𝐴P) → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
236, 1, 22syl2anc 397 . . . . . . . . . 10 (𝜑 → (𝐵 ·P 𝐴) = (𝐴 ·P 𝐵))
24 recidpirqlemcalc.rec . . . . . . . . . 10 (𝜑 → (𝐴 ·P 𝐵) = 1P)
2523, 24eqtrd 2086 . . . . . . . . 9 (𝜑 → (𝐵 ·P 𝐴) = 1P)
26 1idpr 6718 . . . . . . . . . 10 (𝐵P → (𝐵 ·P 1P) = 𝐵)
276, 26syl 14 . . . . . . . . 9 (𝜑 → (𝐵 ·P 1P) = 𝐵)
2825, 27oveq12d 5555 . . . . . . . 8 (𝜑 → ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)) = (1P +P 𝐵))
2919, 21, 283eqtrd 2090 . . . . . . 7 (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (1P +P 𝐵))
3029oveq1d 5552 . . . . . 6 (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
3114, 17, 303eqtrd 2090 . . . . 5 (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
32 1idpr 6718 . . . . . 6 (1PP → (1P ·P 1P) = 1P)
332, 32mp1i 10 . . . . 5 (𝜑 → (1P ·P 1P) = 1P)
3431, 33oveq12d 5555 . . . 4 (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P))
35 addcomprg 6704 . . . . . . . 8 ((1PP𝐵P) → (1P +P 𝐵) = (𝐵 +P 1P))
363, 6, 35syl2anc 397 . . . . . . 7 (𝜑 → (1P +P 𝐵) = (𝐵 +P 1P))
3736oveq1d 5552 . . . . . 6 (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐵 +P 1P) +P (𝐴 +P 1P)))
38 addcomprg 6704 . . . . . . 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 397 . . . . . 6 (𝜑 → ((𝐵 +P 1P) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
4037, 39eqtrd 2086 . . . . 5 (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
4140oveq1d 5552 . . . 4 (𝜑 → (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
4234, 41eqtrd 2086 . . 3 (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
4342oveq1d 5552 . 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 6706 . . . . . 6 ((1PP ∧ (𝐵 +P 1P) ∈ P) → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
453, 8, 44syl2anc 397 . . . . 5 (𝜑 → (1P ·P (𝐵 +P 1P)) = ((𝐵 +P 1P) ·P 1P))
46 1idpr 6718 . . . . . 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 2086 . . . 4 (𝜑 → (1P ·P (𝐵 +P 1P)) = (𝐵 +P 1P))
4916, 48oveq12d 5555 . . 3 (𝜑 → (((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
5049oveq1d 5552 . 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 2096 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 1257  wcel 1407  (class class class)co 5537  Pcnp 6417  1Pc1p 6418   +P cpp 6419   ·P cmp 6420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-eprel 4051  df-id 4055  df-po 4058  df-iso 4059  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-1o 6029  df-2o 6030  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-pli 6431  df-mi 6432  df-lti 6433  df-plpq 6470  df-mpq 6471  df-enq 6473  df-nqqs 6474  df-plqqs 6475  df-mqqs 6476  df-1nqqs 6477  df-rq 6478  df-ltnqqs 6479  df-enq0 6550  df-nq0 6551  df-0nq0 6552  df-plq0 6553  df-mq0 6554  df-inp 6592  df-i1p 6593  df-iplp 6594  df-imp 6595
This theorem is referenced by:  recidpirq  6962
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