Theorem asclpropd 15023

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 6106	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ (1r‘𝐾) ∈ 𝑊)) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)))
4	3	anassrs 404	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑃) ∧ (1r‘𝐾) ∈ 𝑊) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)))
5	1, 4	mpidan 427	. . . . 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 6095	. . . . . 6 ⊢ (𝜑 → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
8	7	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
9	5, 8	eqtrd 2271	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
10	9	mpteq2dva 4219	. . 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 4214	. . 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 4214	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))))
15	10, 12, 14	3eqtr3d 2279	. 2 ⊢ (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))))
16		eqid 2238	. . 3 ⊢ (algSc‘𝐾) = (algSc‘𝐾)
17		asclpropd.f	. . 3 ⊢ 𝐹 = (Scalar‘𝐾)
18		eqid 2238	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
19		eqid 2238	. . 3 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾)
20		eqid 2238	. . 3 ⊢ (1r‘𝐾) = (1r‘𝐾)
21	16, 17, 18, 19, 20	asclfval 15004	. 2 ⊢ (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)))
22		eqid 2238	. . 3 ⊢ (algSc‘𝐿) = (algSc‘𝐿)
23		asclpropd.g	. . 3 ⊢ 𝐺 = (Scalar‘𝐿)
24		eqid 2238	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
25		eqid 2238	. . 3 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿)
26		eqid 2238	. . 3 ⊢ (1r‘𝐿) = (1r‘𝐿)
27	22, 23, 24, 25, 26	asclfval 15004	. 2 ⊢ (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
28	15, 21, 27	3eqtr4g 2296	1 ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209   ↦ cmpt 4190  ‘cfv 5375  (class class class)co 6079  Basecbs 13335  Scalarcsca 13417   ·_𝑠 cvsca 13418  1_rcur 14245  algSccascl 14981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-ndx 13338  df-slot 13339  df-base 13341  df-sca 13430  df-ascl 14984

This theorem is referenced by: (None)