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 1_r 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	⊢ (𝜑 → 𝑃 = (Base‘𝐹))
asclpropd.2	⊢ (𝜑 → 𝑃 = (Base‘𝐺))
asclpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑊)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
asclpropd.4	⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿))
asclpropd.5	⊢ (𝜑 → (1r‘𝐾) ∈ 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.

Step	Hyp	Ref	Expression
1		asclpropd.5	. . . . . 6 ⊢ (𝜑 → (1r‘𝐾) ∈ 𝑊)
2		asclpropd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑊)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
3	2	oveqrspc2v 7191	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ (1r‘𝐾) ∈ 𝑊)) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)))
4	3	anassrs 471	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑃) ∧ (1r‘𝐾) ∈ 𝑊) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)))
5	1, 4	mpidan 689	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)))
6		asclpropd.4	. . . . . . 7 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿))
7	6	oveq2d 7180	. . . . . 6 ⊢ (𝜑 → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
8	7	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
9	5, 8	eqtrd 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
10	9	mpteq2dva 5122	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) = (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))))
11		asclpropd.1	. . . 4 ⊢ (𝜑 → 𝑃 = (Base‘𝐹))
12	11	mpteq1d 5116	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))))
13		asclpropd.2	. . . 4 ⊢ (𝜑 → 𝑃 = (Base‘𝐺))
14	13	mpteq1d 5116	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))))
15	10, 12, 14	3eqtr3d 2781	. 2 ⊢ (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))))
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 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
20		eqid 2738	. . 3 ⊢ (1r‘𝐾) = (1r‘𝐾)
21	16, 17, 18, 19, 20	asclfval 20685	. 2 ⊢ (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)))
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 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
26		eqid 2738	. . 3 ⊢ (1r‘𝐿) = (1r‘𝐿)
27	22, 23, 24, 25, 26	asclfval 20685	. 2 ⊢ (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
28	15, 21, 27	3eqtr4g 2798	1 ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))

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  1_rcur 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