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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 7191 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ (1r‘𝐾) ∈ 𝑊)) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾))) |
4 | 3 | anassrs 471 | . . . . . 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 7180 | . . . . . 6 ⊢ (𝜑 → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
8 | 7 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
9 | 5, 8 | eqtrd 2773 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
10 | 9 | mpteq2dva 5122 | . . 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 5116 | . . 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 5116 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))) |
15 | 10, 12, 14 | 3eqtr3d 2781 | . 2 ⊢ (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))) |
16 | eqid 2738 | . . 3 ⊢ (algSc‘𝐾) = (algSc‘𝐾) | |
17 | asclpropd.f | . . 3 ⊢ 𝐹 = (Scalar‘𝐾) | |
18 | eqid 2738 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
19 | eqid 2738 | . . 3 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾) | |
20 | eqid 2738 | . . 3 ⊢ (1r‘𝐾) = (1r‘𝐾) | |
21 | 16, 17, 18, 19, 20 | asclfval 20685 | . 2 ⊢ (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) |
22 | eqid 2738 | . . 3 ⊢ (algSc‘𝐿) = (algSc‘𝐿) | |
23 | asclpropd.g | . . 3 ⊢ 𝐺 = (Scalar‘𝐿) | |
24 | eqid 2738 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
25 | eqid 2738 | . . 3 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿) | |
26 | eqid 2738 | . . 3 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
27 | 22, 23, 24, 25, 26 | asclfval 20685 | . 2 ⊢ (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
28 | 15, 21, 27 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ↦ cmpt 5107 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 Scalarcsca 16664 ·𝑠 cvsca 16665 1rcur 19363 algSccascl 20661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-slot 16583 df-base 16585 df-ascl 20664 |
This theorem is referenced by: ply1ascl 21026 |
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