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Theorem asclpropd 19569
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 6838 . . . . . . 7 ((𝜑 ∧ (𝑧𝑃 ∧ (1r𝐾) ∈ 𝑊)) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
43anassrs 683 . . . . . 6 (((𝜑𝑧𝑃) ∧ (1r𝐾) ∈ 𝑊) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
51, 4mpidan 707 . . . . 5 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
6 asclpropd.4 . . . . . . 7 (𝜑 → (1r𝐾) = (1r𝐿))
76oveq2d 6831 . . . . . 6 (𝜑 → (𝑧( ·𝑠𝐿)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
87adantr 472 . . . . 5 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐿)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
95, 8eqtrd 2795 . . . 4 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
109mpteq2dva 4897 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧𝑃 ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
11 asclpropd.1 . . . 4 (𝜑𝑃 = (Base‘𝐹))
1211mpteq1d 4891 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾))))
13 asclpropd.2 . . . 4 (𝜑𝑃 = (Base‘𝐺))
1413mpteq1d 4891 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐿)(1r𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
1510, 12, 143eqtr3d 2803 . 2 (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
16 eqid 2761 . . 3 (algSc‘𝐾) = (algSc‘𝐾)
17 asclpropd.f . . 3 𝐹 = (Scalar‘𝐾)
18 eqid 2761 . . 3 (Base‘𝐹) = (Base‘𝐹)
19 eqid 2761 . . 3 ( ·𝑠𝐾) = ( ·𝑠𝐾)
20 eqid 2761 . . 3 (1r𝐾) = (1r𝐾)
2116, 17, 18, 19, 20asclfval 19557 . 2 (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾)))
22 eqid 2761 . . 3 (algSc‘𝐿) = (algSc‘𝐿)
23 asclpropd.g . . 3 𝐺 = (Scalar‘𝐿)
24 eqid 2761 . . 3 (Base‘𝐺) = (Base‘𝐺)
25 eqid 2761 . . 3 ( ·𝑠𝐿) = ( ·𝑠𝐿)
26 eqid 2761 . . 3 (1r𝐿) = (1r𝐿)
2722, 23, 24, 25, 26asclfval 19557 . 2 (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿)))
2815, 21, 273eqtr4g 2820 1 (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2140  cmpt 4882  cfv 6050  (class class class)co 6815  Basecbs 16080  Scalarcsca 16167   ·𝑠 cvsca 16168  1rcur 18722  algSccascl 19534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-slot 16084  df-base 16086  df-ascl 19537
This theorem is referenced by:  ply1ascl  19851
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