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Theorem genpassg 6581
Description: Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.)
Hypotheses
Ref Expression
genpelvl.1 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
genpassg.4 dom 𝐹 = (P × P)
genpassg.5 ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)
genpassg.6 ((𝑓Q𝑔QQ) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))
Assertion
Ref Expression
genpassg ((𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝑔,,𝑤,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓,𝑔,,𝑤,𝑣   𝑥,𝐺,𝑦,𝑧,𝑓,𝑔,,𝑤,𝑣   𝑓,𝐹,𝑔   𝐶,𝑓,𝑔,,𝑣,𝑤,𝑥,𝑦,𝑧   ,𝐹,𝑣,𝑤,𝑥,𝑦,𝑧

Proof of Theorem genpassg
StepHypRef Expression
1 genpelvl.1 . . 3 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
2 genpelvl.2 . . 3 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
3 genpassg.4 . . 3 dom 𝐹 = (P × P)
4 genpassg.5 . . 3 ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)
5 genpassg.6 . . 3 ((𝑓Q𝑔QQ) → ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺)))
61, 2, 3, 4, 5genpassl 6579 . 2 ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))))
71, 2, 3, 4, 5genpassu 6580 . 2 ((𝐴P𝐵P𝐶P) → (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))
84caovcl 5618 . . . . 5 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ P)
94caovcl 5618 . . . . 5 (((𝐴𝐹𝐵) ∈ P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) ∈ P)
108, 9sylan 267 . . . 4 (((𝐴P𝐵P) ∧ 𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) ∈ P)
11103impa 1099 . . 3 ((𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) ∈ P)
124caovcl 5618 . . . . 5 ((𝐵P𝐶P) → (𝐵𝐹𝐶) ∈ P)
134caovcl 5618 . . . . 5 ((𝐴P ∧ (𝐵𝐹𝐶) ∈ P) → (𝐴𝐹(𝐵𝐹𝐶)) ∈ P)
1412, 13sylan2 270 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (𝐴𝐹(𝐵𝐹𝐶)) ∈ P)
15143impb 1100 . . 3 ((𝐴P𝐵P𝐶P) → (𝐴𝐹(𝐵𝐹𝐶)) ∈ P)
16 preqlu 6527 . . 3 ((((𝐴𝐹𝐵)𝐹𝐶) ∈ P ∧ (𝐴𝐹(𝐵𝐹𝐶)) ∈ P) → (((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) ↔ ((1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))) ∧ (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))))
1711, 15, 16syl2anc 391 . 2 ((𝐴P𝐵P𝐶P) → (((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) ↔ ((1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))) ∧ (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))))
186, 7, 17mpbir2and 851 1 ((𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  wrex 2304  {crab 2307  cop 3375   × cxp 4306  dom cdm 4308  cfv 4865  (class class class)co 5475  cmpt2 5477  1st c1st 5728  2nd c2nd 5729  Qcnq 6335  Pcnp 6346
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 3869  ax-sep 3872  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274
This theorem depends on definitions:  df-bi 110  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 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-id 4027  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-qs 6075  df-ni 6359  df-nqqs 6403  df-inp 6521
This theorem is referenced by:  addassprg  6634  mulassprg  6636
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