Theorem asclpropd 21855

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 7430	. . . . . . 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 7419	. . . . . 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 2770	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
10	9	mpteq2dva 5214	. . 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 5210	. . 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 5210	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))))
15	10, 12, 14	3eqtr3d 2778	. 2 ⊢ (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))))
16		eqid 2735	. . 3 ⊢ (algSc‘𝐾) = (algSc‘𝐾)
17		asclpropd.f	. . 3 ⊢ 𝐹 = (Scalar‘𝐾)
18		eqid 2735	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
19		eqid 2735	. . 3 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
20		eqid 2735	. . 3 ⊢ (1r‘𝐾) = (1r‘𝐾)
21	16, 17, 18, 19, 20	asclfval 21837	. 2 ⊢ (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)))
22		eqid 2735	. . 3 ⊢ (algSc‘𝐿) = (algSc‘𝐿)
23		asclpropd.g	. . 3 ⊢ 𝐺 = (Scalar‘𝐿)
24		eqid 2735	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
25		eqid 2735	. . 3 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
26		eqid 2735	. . 3 ⊢ (1r‘𝐿) = (1r‘𝐿)
27	22, 23, 24, 25, 26	asclfval 21837	. 2 ⊢ (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
28	15, 21, 27	3eqtr4g 2795	1 ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ↦ cmpt 5201  ‘cfv 6530  (class class class)co 7403  Basecbs 17226  Scalarcsca 17272   ·_𝑠 cvsca 17273  1_rcur 20139  algSccascl 21810

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-1cn 11185  ax-addcl 11187

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-om 7860  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-nn 12239  df-slot 17199  df-ndx 17211  df-base 17227  df-ascl 21813

This theorem is referenced by:  ply1ascl  22193