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Theorem recidpirqlemcalc 7338
Description: Lemma for recidpirq 7339. 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 7057 . . . . . 6 1PP
32a1i 9 . . . . 5 (𝜑 → 1PP)
4 addclpr 7040 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
51, 3, 4syl2anc 403 . . . 4 (𝜑 → (𝐴 +P 1P) ∈ P)
6 recidpirqlemcalc.b . . . . 5 (𝜑𝐵P)
7 addclpr 7040 . . . . 5 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
86, 3, 7syl2anc 403 . . . 4 (𝜑 → (𝐵 +P 1P) ∈ P)
9 addclpr 7040 . . . 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 7082 . . 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 1172 . 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 7091 . . . . . . 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 1172 . . . . . 6 (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)))
15 1idpr 7095 . . . . . . . 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 5629 . . . . . 6 (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P ((𝐴 +P 1P) ·P 1P)) = (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)))
18 mulcomprg 7083 . . . . . . . . 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 7091 . . . . . . . . 9 ((𝐵P𝐴P ∧ 1PP) → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
216, 1, 3, 20syl3anc 1172 . . . . . . . 8 (𝜑 → (𝐵 ·P (𝐴 +P 1P)) = ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)))
22 mulcomprg 7083 . . . . . . . . . . 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 2117 . . . . . . . . 9 (𝜑 → (𝐵 ·P 𝐴) = 1P)
26 1idpr 7095 . . . . . . . . . 10 (𝐵P → (𝐵 ·P 1P) = 𝐵)
276, 26syl 14 . . . . . . . . 9 (𝜑 → (𝐵 ·P 1P) = 𝐵)
2825, 27oveq12d 5631 . . . . . . . 8 (𝜑 → ((𝐵 ·P 𝐴) +P (𝐵 ·P 1P)) = (1P +P 𝐵))
2919, 21, 283eqtrd 2121 . . . . . . 7 (𝜑 → ((𝐴 +P 1P) ·P 𝐵) = (1P +P 𝐵))
3029oveq1d 5628 . . . . . 6 (𝜑 → (((𝐴 +P 1P) ·P 𝐵) +P (𝐴 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
3114, 17, 303eqtrd 2121 . . . . 5 (𝜑 → ((𝐴 +P 1P) ·P (𝐵 +P 1P)) = ((1P +P 𝐵) +P (𝐴 +P 1P)))
32 1idpr 7095 . . . . . 6 (1PP → (1P ·P 1P) = 1P)
332, 32mp1i 10 . . . . 5 (𝜑 → (1P ·P 1P) = 1P)
3431, 33oveq12d 5631 . . . 4 (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P))
35 addcomprg 7081 . . . . . . . 8 ((1PP𝐵P) → (1P +P 𝐵) = (𝐵 +P 1P))
363, 6, 35syl2anc 403 . . . . . . 7 (𝜑 → (1P +P 𝐵) = (𝐵 +P 1P))
3736oveq1d 5628 . . . . . 6 (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐵 +P 1P) +P (𝐴 +P 1P)))
38 addcomprg 7081 . . . . . . 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 2117 . . . . 5 (𝜑 → ((1P +P 𝐵) +P (𝐴 +P 1P)) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
4140oveq1d 5628 . . . 4 (𝜑 → (((1P +P 𝐵) +P (𝐴 +P 1P)) +P 1P) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
4234, 41eqtrd 2117 . . 3 (𝜑 → (((𝐴 +P 1P) ·P (𝐵 +P 1P)) +P (1P ·P 1P)) = (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P))
4342oveq1d 5628 . 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 7083 . . . . . 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 7095 . . . . . 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 2117 . . . 4 (𝜑 → (1P ·P (𝐵 +P 1P)) = (𝐵 +P 1P))
4916, 48oveq12d 5631 . . 3 (𝜑 → (((𝐴 +P 1P) ·P 1P) +P (1P ·P (𝐵 +P 1P))) = ((𝐴 +P 1P) +P (𝐵 +P 1P)))
5049oveq1d 5628 . 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 2127 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 1287  wcel 1436  (class class class)co 5613  Pcnp 6794  1Pc1p 6795   +P cpp 6796   ·P cmp 6797
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-eprel 4090  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-1o 6135  df-2o 6136  df-oadd 6139  df-omul 6140  df-er 6244  df-ec 6246  df-qs 6250  df-ni 6807  df-pli 6808  df-mi 6809  df-lti 6810  df-plpq 6847  df-mpq 6848  df-enq 6850  df-nqqs 6851  df-plqqs 6852  df-mqqs 6853  df-1nqqs 6854  df-rq 6855  df-ltnqqs 6856  df-enq0 6927  df-nq0 6928  df-0nq0 6929  df-plq0 6930  df-mq0 6931  df-inp 6969  df-i1p 6970  df-iplp 6971  df-imp 6972
This theorem is referenced by:  recidpirq  7339
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