MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  asclpropd Structured version   Visualization version   GIF version

Theorem asclpropd 21934
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 7457 . . . . . . 7 ((𝜑 ∧ (𝑧𝑃 ∧ (1r𝐾) ∈ 𝑊)) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐾)))
43anassrs 467 . . . . . 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 7446 . . . . . 6 (𝜑 → (𝑧( ·𝑠𝐿)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
87adantr 480 . . . . 5 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐿)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
95, 8eqtrd 2774 . . . 4 ((𝜑𝑧𝑃) → (𝑧( ·𝑠𝐾)(1r𝐾)) = (𝑧( ·𝑠𝐿)(1r𝐿)))
109mpteq2dva 5247 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧𝑃 ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
11 asclpropd.1 . . . 4 (𝜑𝑃 = (Base‘𝐹))
1211mpteq1d 5242 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾))))
13 asclpropd.2 . . . 4 (𝜑𝑃 = (Base‘𝐺))
1413mpteq1d 5242 . . 3 (𝜑 → (𝑧𝑃 ↦ (𝑧( ·𝑠𝐿)(1r𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
1510, 12, 143eqtr3d 2782 . 2 (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿))))
16 eqid 2734 . . 3 (algSc‘𝐾) = (algSc‘𝐾)
17 asclpropd.f . . 3 𝐹 = (Scalar‘𝐾)
18 eqid 2734 . . 3 (Base‘𝐹) = (Base‘𝐹)
19 eqid 2734 . . 3 ( ·𝑠𝐾) = ( ·𝑠𝐾)
20 eqid 2734 . . 3 (1r𝐾) = (1r𝐾)
2116, 17, 18, 19, 20asclfval 21916 . 2 (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠𝐾)(1r𝐾)))
22 eqid 2734 . . 3 (algSc‘𝐿) = (algSc‘𝐿)
23 asclpropd.g . . 3 𝐺 = (Scalar‘𝐿)
24 eqid 2734 . . 3 (Base‘𝐺) = (Base‘𝐺)
25 eqid 2734 . . 3 ( ·𝑠𝐿) = ( ·𝑠𝐿)
26 eqid 2734 . . 3 (1r𝐿) = (1r𝐿)
2722, 23, 24, 25, 26asclfval 21916 . 2 (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠𝐿)(1r𝐿)))
2815, 21, 273eqtr4g 2799 1 (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  cmpt 5230  cfv 6562  (class class class)co 7430  Basecbs 17244  Scalarcsca 17300   ·𝑠 cvsca 17301  1rcur 20198  algSccascl 21889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-1cn 11210  ax-addcl 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-nn 12264  df-slot 17215  df-ndx 17227  df-base 17245  df-ascl 21892
This theorem is referenced by:  ply1ascl  22276
  Copyright terms: Public domain W3C validator