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Theorem genpassg 7346
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 7344 . 2 ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))))
71, 2, 3, 4, 5genpassu 7345 . 2 ((𝐴P𝐵P𝐶P) → (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))
84caovcl 5925 . . . . 5 ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ P)
94caovcl 5925 . . . . 5 (((𝐴𝐹𝐵) ∈ P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) ∈ P)
108, 9sylan 281 . . . 4 (((𝐴P𝐵P) ∧ 𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) ∈ P)
11103impa 1176 . . 3 ((𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) ∈ P)
124caovcl 5925 . . . . 5 ((𝐵P𝐶P) → (𝐵𝐹𝐶) ∈ P)
134caovcl 5925 . . . . 5 ((𝐴P ∧ (𝐵𝐹𝐶) ∈ P) → (𝐴𝐹(𝐵𝐹𝐶)) ∈ P)
1412, 13sylan2 284 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (𝐴𝐹(𝐵𝐹𝐶)) ∈ P)
15143impb 1177 . . 3 ((𝐴P𝐵P𝐶P) → (𝐴𝐹(𝐵𝐹𝐶)) ∈ P)
16 preqlu 7292 . . 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 408 . 2 ((𝐴P𝐵P𝐶P) → (((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) ↔ ((1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))) ∧ (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶))))))
186, 7, 17mpbir2and 928 1 ((𝐴P𝐵P𝐶P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  wrex 2417  {crab 2420  cop 3530   × cxp 4537  dom cdm 4539  cfv 5123  (class class class)co 5774  cmpo 5776  1st c1st 6036  2nd c2nd 6037  Qcnq 7100  Pcnp 7111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7124  df-nqqs 7168  df-inp 7286
This theorem is referenced by:  addassprg  7399  mulassprg  7401
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