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Mirrors > Home > MPE Home > Th. List > asclpropd | Structured version Visualization version GIF version |
Description: If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on 1r 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.) |
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) |
Ref | Expression |
---|---|
asclpropd | ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿)) |
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 7182 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ (1r‘𝐾) ∈ 𝑊)) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾))) |
4 | 3 | anassrs 470 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑃) ∧ (1r‘𝐾) ∈ 𝑊) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾))) |
5 | 1, 4 | mpidan 687 | . . . . 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 7171 | . . . . . 6 ⊢ (𝜑 → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
8 | 7 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
9 | 5, 8 | eqtrd 2856 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
10 | 9 | mpteq2dva 5160 | . . 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 5154 | . . 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 5154 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))) |
15 | 10, 12, 14 | 3eqtr3d 2864 | . 2 ⊢ (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))) |
16 | eqid 2821 | . . 3 ⊢ (algSc‘𝐾) = (algSc‘𝐾) | |
17 | asclpropd.f | . . 3 ⊢ 𝐹 = (Scalar‘𝐾) | |
18 | eqid 2821 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
19 | eqid 2821 | . . 3 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾) | |
20 | eqid 2821 | . . 3 ⊢ (1r‘𝐾) = (1r‘𝐾) | |
21 | 16, 17, 18, 19, 20 | asclfval 20107 | . 2 ⊢ (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) |
22 | eqid 2821 | . . 3 ⊢ (algSc‘𝐿) = (algSc‘𝐿) | |
23 | asclpropd.g | . . 3 ⊢ 𝐺 = (Scalar‘𝐿) | |
24 | eqid 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
25 | eqid 2821 | . . 3 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿) | |
26 | eqid 2821 | . . 3 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
27 | 22, 23, 24, 25, 26 | asclfval 20107 | . 2 ⊢ (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
28 | 15, 21, 27 | 3eqtr4g 2881 | 1 ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 Scalarcsca 16567 ·𝑠 cvsca 16568 1rcur 19250 algSccascl 20083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-slot 16486 df-base 16488 df-ascl 20086 |
This theorem is referenced by: ply1ascl 20425 |
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