Theorem asclpropd 21887

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 7387	. . . . . . 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 690	. . . . 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 7376	. . . . . 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 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
10	9	mpteq2dva 5179	. . 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 5176	. . 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 5176	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))))
15	10, 12, 14	3eqtr3d 2780	. 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 21868	. 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 21868	. 2 ⊢ (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))
28	15, 21, 27	3eqtr4g 2797	1 ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ↦ cmpt 5167  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  Scalarcsca 17214   ·_𝑠 cvsca 17215  1_rcur 20153  algSccascl 21842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-1cn 11087  ax-addcl 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-nn 12166  df-slot 17143  df-ndx 17155  df-base 17171  df-ascl 21845

This theorem is referenced by:  ply1ascl  22233