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Theorem asclpropd 20054
Description: If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on 1r can be discharged either by letting 𝑊 = V (if strong equality is known on ·𝑠) or assuming 𝐾 is a ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
Hypotheses
Ref Expression
asclpropd.f 𝐹 = (Scalar‘𝐾)
asclpropd.g 𝐺 = (Scalar‘𝐿)
asclpropd.1 (𝜑𝑃 = (Base‘𝐹))
asclpropd.2 (𝜑𝑃 = (Base‘𝐺))
asclpropd.3 ((𝜑 ∧ (𝑥𝑃𝑦𝑊)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
asclpropd.4 (𝜑 → (1r𝐾) = (1r𝐿))
asclpropd.5 (𝜑 → (1r𝐾) ∈ 𝑊)
Assertion
Ref Expression
asclpropd (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))
Distinct variable groups:   𝑥,𝑦,𝐾   𝑥,𝐿,𝑦   𝑥,𝑃,𝑦   𝜑,𝑥,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem asclpropd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 asclpropd.5 . . . . . 6 (𝜑 → (1r𝐾) ∈ 𝑊)
2 asclpropd.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑃𝑦𝑊)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
32oveqrspc2v 7172 . . . . . . 7 ((𝜑 ∧ (𝑧𝑃 ∧ (1r𝐾) ∈ 𝑊)) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
43anassrs 468 . . . . . 6 (((𝜑𝑧𝑃) ∧ (1r𝐾) ∈ 𝑊) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
51, 4mpidan 685 . . . . 5 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
6 asclpropd.4 . . . . . . 7 (𝜑 → (1r𝐾) = (1r𝐿))
76oveq2d 7161 . . . . . 6 (𝜑 → (𝑧( ·𝑠𝐿)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
87adantr 481 . . . . 5 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐿)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
95, 8eqtrd 2853 . . . 4 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
109mpteq2dva 5152 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧𝑃 ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
11 asclpropd.1 . . . 4 (𝜑𝑃 = (Base‘𝐹))
1211mpteq1d 5146 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾))))
13 asclpropd.2 . . . 4 (𝜑𝑃 = (Base‘𝐺))
1413mpteq1d 5146 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐿)(1r𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
1510, 12, 143eqtr3d 2861 . 2 (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
16 eqid 2818 . . 3 (algSc‘𝐾) = (algSc‘𝐾)
17 asclpropd.f . . 3 𝐹 = (Scalar‘𝐾)
18 eqid 2818 . . 3 (Base‘𝐹) = (Base‘𝐹)
19 eqid 2818 . . 3 ( ·𝑠𝐾) = ( ·𝑠𝐾)
20 eqid 2818 . . 3 (1r𝐾) = (1r𝐾)
2116, 17, 18, 19, 20asclfval 20036 . 2 (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾)))
22 eqid 2818 . . 3 (algSc‘𝐿) = (algSc‘𝐿)
23 asclpropd.g . . 3 𝐺 = (Scalar‘𝐿)
24 eqid 2818 . . 3 (Base‘𝐺) = (Base‘𝐺)
25 eqid 2818 . . 3 ( ·𝑠𝐿) = ( ·𝑠𝐿)
26 eqid 2818 . . 3 (1r𝐿) = (1r𝐿)
2722, 23, 24, 25, 26asclfval 20036 . 2 (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿)))
2815, 21, 273eqtr4g 2878 1 (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cmpt 5137  cfv 6348  (class class class)co 7145  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557  1rcur 19180  algSccascl 20012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-slot 16475  df-base 16477  df-ascl 20015
This theorem is referenced by:  ply1ascl  20354
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