Theorem asclpropd 21917

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 7458	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ (1r‘𝐾) ∈ 𝑊)) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)))
4	3	anassrs 467	. . . . . 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 7447	. . . . . 6 ⊢ (𝜑 → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
9	5, 8	eqtrd 2777	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
10	9	mpteq2dva 5242	. . 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 5237	. . 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 5237	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))))
15	10, 12, 14	3eqtr3d 2785	. 2 ⊢ (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))))
16		eqid 2737	. . 3 ⊢ (algSc‘𝐾) = (algSc‘𝐾)
17		asclpropd.f	. . 3 ⊢ 𝐹 = (Scalar‘𝐾)
18		eqid 2737	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
19		eqid 2737	. . 3 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
20		eqid 2737	. . 3 ⊢ (1r‘𝐾) = (1r‘𝐾)
21	16, 17, 18, 19, 20	asclfval 21899	. 2 ⊢ (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)))
22		eqid 2737	. . 3 ⊢ (algSc‘𝐿) = (algSc‘𝐿)
23		asclpropd.g	. . 3 ⊢ 𝐺 = (Scalar‘𝐿)
24		eqid 2737	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
25		eqid 2737	. . 3 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
26		eqid 2737	. . 3 ⊢ (1r‘𝐿) = (1r‘𝐿)
27	22, 23, 24, 25, 26	asclfval 21899	. 2 ⊢ (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
28	15, 21, 27	3eqtr4g 2802	1 ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ↦ cmpt 5225  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  Scalarcsca 17300   ·_𝑠 cvsca 17301  1_rcur 20178  algSccascl 21872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-slot 17219  df-ndx 17231  df-base 17248  df-ascl 21875

This theorem is referenced by:  ply1ascl  22261