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Theorem caovlem2d 5770
Description: Rearrangement of expression involving multiplication (𝐺) and addition (𝐹). (Contributed by Jim Kingdon, 3-Jan-2020.)
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)
caovdl2.6 (𝜑𝑅𝑆)
caovdl2.com ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
caovdl2.ass ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
caovdl2.cl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
Assertion
Ref Expression
caovlem2d (𝜑 → ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(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,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovlem2d
Dummy variables 𝑠 𝑟 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caovdilemd.cl . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆)
2 caovdilemd.a . . . 4 (𝜑𝐴𝑆)
3 caovdilemd.c . . . . 5 (𝜑𝐶𝑆)
4 caovdilemd.h . . . . 5 (𝜑𝐻𝑆)
51, 3, 4caovcld 5731 . . . 4 (𝜑 → (𝐶𝐺𝐻) ∈ 𝑆)
61, 2, 5caovcld 5731 . . 3 (𝜑 → (𝐴𝐺(𝐶𝐺𝐻)) ∈ 𝑆)
7 caovdilemd.b . . . 4 (𝜑𝐵𝑆)
8 caovdilemd.d . . . . 5 (𝜑𝐷𝑆)
91, 8, 4caovcld 5731 . . . 4 (𝜑 → (𝐷𝐺𝐻) ∈ 𝑆)
101, 7, 9caovcld 5731 . . 3 (𝜑 → (𝐵𝐺(𝐷𝐺𝐻)) ∈ 𝑆)
11 caovdl2.6 . . . . 5 (𝜑𝑅𝑆)
121, 8, 11caovcld 5731 . . . 4 (𝜑 → (𝐷𝐺𝑅) ∈ 𝑆)
131, 2, 12caovcld 5731 . . 3 (𝜑 → (𝐴𝐺(𝐷𝐺𝑅)) ∈ 𝑆)
14 caovdl2.com . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
15 caovdl2.ass . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
161, 3, 11caovcld 5731 . . . 4 (𝜑 → (𝐶𝐺𝑅) ∈ 𝑆)
171, 7, 16caovcld 5731 . . 3 (𝜑 → (𝐵𝐺(𝐶𝐺𝑅)) ∈ 𝑆)
18 caovdl2.cl . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
196, 10, 13, 14, 15, 17, 18caov42d 5764 . 2 (𝜑 → (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))𝐹((𝐴𝐺(𝐷𝐺𝑅))𝐹(𝐵𝐺(𝐶𝐺𝑅)))) = (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐴𝐺(𝐷𝐺𝑅)))𝐹((𝐵𝐺(𝐶𝐺𝑅))𝐹(𝐵𝐺(𝐷𝐺𝐻)))))
20 caovdilemd.com . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
21 caovdilemd.distr . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧)))
22 caovdilemd.ass . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
2320, 21, 22, 1, 2, 7, 3, 8, 4caovdilemd 5769 . . 3 (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
2420, 21, 22, 1, 2, 7, 8, 3, 11caovdilemd 5769 . . 3 (𝜑 → (((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅) = ((𝐴𝐺(𝐷𝐺𝑅))𝐹(𝐵𝐺(𝐶𝐺𝑅))))
2523, 24oveq12d 5607 . 2 (𝜑 → ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))𝐹((𝐴𝐺(𝐷𝐺𝑅))𝐹(𝐵𝐺(𝐶𝐺𝑅)))))
26 simpr1 945 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → 𝑥𝑆)
2718caovclg 5730 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑆𝑠𝑆)) → (𝑟𝐹𝑠) ∈ 𝑆)
2827caovclg 5730 . . . . . . . 8 ((𝜑 ∧ (𝑦𝑆𝑧𝑆)) → (𝑦𝐹𝑧) ∈ 𝑆)
29283adantr1 1098 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑦𝐹𝑧) ∈ 𝑆)
3026, 29jca 300 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑆 ∧ (𝑦𝐹𝑧) ∈ 𝑆))
3120caovcomg 5733 . . . . . . 7 ((𝜑 ∧ (𝑟𝑆𝑠𝑆)) → (𝑟𝐺𝑠) = (𝑠𝐺𝑟))
3231caovcomg 5733 . . . . . 6 ((𝜑 ∧ (𝑥𝑆 ∧ (𝑦𝐹𝑧) ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑦𝐹𝑧)𝐺𝑥))
3330, 32syldan 276 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑦𝐹𝑧)𝐺𝑥))
34 3anrot 925 . . . . . 6 ((𝑥𝑆𝑦𝑆𝑧𝑆) ↔ (𝑦𝑆𝑧𝑆𝑥𝑆))
3521caovdirg 5755 . . . . . . 7 ((𝜑 ∧ (𝑟𝑆𝑠𝑆𝑡𝑆)) → ((𝑟𝐹𝑠)𝐺𝑡) = ((𝑟𝐺𝑡)𝐹(𝑠𝐺𝑡)))
3635caovdirg 5755 . . . . . 6 ((𝜑 ∧ (𝑦𝑆𝑧𝑆𝑥𝑆)) → ((𝑦𝐹𝑧)𝐺𝑥) = ((𝑦𝐺𝑥)𝐹(𝑧𝐺𝑥)))
3734, 36sylan2b 281 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑦𝐹𝑧)𝐺𝑥) = ((𝑦𝐺𝑥)𝐹(𝑧𝐺𝑥)))
3820eqcomd 2088 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑦𝐺𝑥) = (𝑥𝐺𝑦))
39383adantr3 1100 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑦𝐺𝑥) = (𝑥𝐺𝑦))
4031caovcomg 5733 . . . . . . . 8 ((𝜑 ∧ (𝑧𝑆𝑥𝑆)) → (𝑧𝐺𝑥) = (𝑥𝐺𝑧))
4140ancom2s 531 . . . . . . 7 ((𝜑 ∧ (𝑥𝑆𝑧𝑆)) → (𝑧𝐺𝑥) = (𝑥𝐺𝑧))
42413adantr2 1099 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑧𝐺𝑥) = (𝑥𝐺𝑧))
4339, 42oveq12d 5607 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑦𝐺𝑥)𝐹(𝑧𝐺𝑥)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
4433, 37, 433eqtrd 2119 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
4544, 2, 5, 12caovdid 5753 . . 3 (𝜑 → (𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅))) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐴𝐺(𝐷𝐺𝑅))))
4644, 7, 16, 9caovdid 5753 . . 3 (𝜑 → (𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻))) = ((𝐵𝐺(𝐶𝐺𝑅))𝐹(𝐵𝐺(𝐷𝐺𝐻))))
4745, 46oveq12d 5607 . 2 (𝜑 → ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻)))) = (((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐴𝐺(𝐷𝐺𝑅)))𝐹((𝐵𝐺(𝐶𝐺𝑅))𝐹(𝐵𝐺(𝐷𝐺𝐻)))))
4819, 25, 473eqtr4d 2125 1 (𝜑 → ((((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻)𝐹(((𝐴𝐺𝐷)𝐹(𝐵𝐺𝐶))𝐺𝑅)) = ((𝐴𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(𝐵𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  (class class class)co 5589
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 4932  df-fv 4975  df-ov 5592
This theorem is referenced by:  mulasssrg  7205
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