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Theorem caovdilemd 5771
Description: Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.)
Hypotheses
Ref Expression
caovdilemd.com ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
caovdilemd.distr ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧)))
caovdilemd.ass ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
caovdilemd.cl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)
caovdilemd.a (𝜑𝐴𝑆)
caovdilemd.b (𝜑𝐵𝑆)
caovdilemd.c (𝜑𝐶𝑆)
caovdilemd.d (𝜑𝐷𝑆)
caovdilemd.h (𝜑𝐻𝑆)
Assertion
Ref Expression
caovdilemd (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovdilemd
StepHypRef Expression
1 caovdilemd.distr . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧)))
2 caovdilemd.cl . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)
3 caovdilemd.a . . . 4 (𝜑𝐴𝑆)
4 caovdilemd.c . . . 4 (𝜑𝐶𝑆)
52, 3, 4caovcld 5733 . . 3 (𝜑 → (𝐴𝐺𝐶) ∈ 𝑆)
6 caovdilemd.b . . . 4 (𝜑𝐵𝑆)
7 caovdilemd.d . . . 4 (𝜑𝐷𝑆)
82, 6, 7caovcld 5733 . . 3 (𝜑 → (𝐵𝐺𝐷) ∈ 𝑆)
9 caovdilemd.h . . 3 (𝜑𝐻𝑆)
101, 5, 8, 9caovdird 5758 . 2 (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)))
11 caovdilemd.ass . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
1211, 3, 4, 9caovassd 5739 . . 3 (𝜑 → ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻)))
1311, 6, 7, 9caovassd 5739 . . 3 (𝜑 → ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻)))
1412, 13oveq12d 5609 . 2 (𝜑 → (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
1510, 14eqtrd 2115 1 (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  (class class class)co 5591
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-iota 4934  df-fv 4977  df-ov 5594
This theorem is referenced by:  caovlem2d  5772  addassnqg  6844  addassnq0  6924  axmulass  7311
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