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Theorem asclpropd 19286
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 6638 . . . . . . 7 ((𝜑 ∧ (𝑧𝑃 ∧ (1r𝐾) ∈ 𝑊)) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
43anassrs 679 . . . . . 6 (((𝜑𝑧𝑃) ∧ (1r𝐾) ∈ 𝑊) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
51, 4mpidan 703 . . . . 5 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
6 asclpropd.4 . . . . . . 7 (𝜑 → (1r𝐾) = (1r𝐿))
76oveq2d 6631 . . . . . 6 (𝜑 → (𝑧( ·𝑠𝐿)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
87adantr 481 . . . . 5 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐿)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
95, 8eqtrd 2655 . . . 4 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
109mpteq2dva 4714 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧𝑃 ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
11 asclpropd.1 . . . 4 (𝜑𝑃 = (Base‘𝐹))
1211mpteq1d 4708 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾))))
13 asclpropd.2 . . . 4 (𝜑𝑃 = (Base‘𝐺))
1413mpteq1d 4708 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐿)(1r𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
1510, 12, 143eqtr3d 2663 . 2 (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
16 eqid 2621 . . 3 (algSc‘𝐾) = (algSc‘𝐾)
17 asclpropd.f . . 3 𝐹 = (Scalar‘𝐾)
18 eqid 2621 . . 3 (Base‘𝐹) = (Base‘𝐹)
19 eqid 2621 . . 3 ( ·𝑠𝐾) = ( ·𝑠𝐾)
20 eqid 2621 . . 3 (1r𝐾) = (1r𝐾)
2116, 17, 18, 19, 20asclfval 19274 . 2 (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾)))
22 eqid 2621 . . 3 (algSc‘𝐿) = (algSc‘𝐿)
23 asclpropd.g . . 3 𝐺 = (Scalar‘𝐿)
24 eqid 2621 . . 3 (Base‘𝐺) = (Base‘𝐺)
25 eqid 2621 . . 3 ( ·𝑠𝐿) = ( ·𝑠𝐿)
26 eqid 2621 . . 3 (1r𝐿) = (1r𝐿)
2722, 23, 24, 25, 26asclfval 19274 . 2 (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿)))
2815, 21, 273eqtr4g 2680 1 (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cmpt 4683  cfv 5857  (class class class)co 6615  Basecbs 15800  Scalarcsca 15884   ·𝑠 cvsca 15885  1rcur 18441  algSccascl 19251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-slot 15804  df-base 15805  df-ascl 19254
This theorem is referenced by:  ply1ascl  19568
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