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Theorem asclpropd 20704
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 7191 . . . . . . 7 ((𝜑 ∧ (𝑧𝑃 ∧ (1r𝐾) ∈ 𝑊)) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
43anassrs 471 . . . . . 6 (((𝜑𝑧𝑃) ∧ (1r𝐾) ∈ 𝑊) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
51, 4mpidan 689 . . . . 5 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
6 asclpropd.4 . . . . . . 7 (𝜑 → (1r𝐾) = (1r𝐿))
76oveq2d 7180 . . . . . 6 (𝜑 → (𝑧( ·𝑠𝐿)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
87adantr 484 . . . . 5 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐿)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
95, 8eqtrd 2773 . . . 4 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
109mpteq2dva 5122 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧𝑃 ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
11 asclpropd.1 . . . 4 (𝜑𝑃 = (Base‘𝐹))
1211mpteq1d 5116 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾))))
13 asclpropd.2 . . . 4 (𝜑𝑃 = (Base‘𝐺))
1413mpteq1d 5116 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐿)(1r𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
1510, 12, 143eqtr3d 2781 . 2 (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
16 eqid 2738 . . 3 (algSc‘𝐾) = (algSc‘𝐾)
17 asclpropd.f . . 3 𝐹 = (Scalar‘𝐾)
18 eqid 2738 . . 3 (Base‘𝐹) = (Base‘𝐹)
19 eqid 2738 . . 3 ( ·𝑠𝐾) = ( ·𝑠𝐾)
20 eqid 2738 . . 3 (1r𝐾) = (1r𝐾)
2116, 17, 18, 19, 20asclfval 20685 . 2 (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾)))
22 eqid 2738 . . 3 (algSc‘𝐿) = (algSc‘𝐿)
23 asclpropd.g . . 3 𝐺 = (Scalar‘𝐿)
24 eqid 2738 . . 3 (Base‘𝐺) = (Base‘𝐺)
25 eqid 2738 . . 3 ( ·𝑠𝐿) = ( ·𝑠𝐿)
26 eqid 2738 . . 3 (1r𝐿) = (1r𝐿)
2722, 23, 24, 25, 26asclfval 20685 . 2 (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿)))
2815, 21, 273eqtr4g 2798 1 (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2113  cmpt 5107  cfv 6333  (class class class)co 7164  Basecbs 16579  Scalarcsca 16664   ·𝑠 cvsca 16665  1rcur 19363  algSccascl 20661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-slot 16583  df-base 16585  df-ascl 20664
This theorem is referenced by:  ply1ascl  21026
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