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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 6572 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ (1r‘𝐾) ∈ 𝑊)) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾))) |
4 | 3 | anassrs 678 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑃) ∧ (1r‘𝐾) ∈ 𝑊) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾))) |
5 | 1, 4 | mpidan 701 | . . . . 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 6565 | . . . . . 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 2644 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
10 | 9 | mpteq2dva 4672 | . . 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 4666 | . . 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 4666 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))) |
15 | 10, 12, 14 | 3eqtr3d 2652 | . 2 ⊢ (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))) |
16 | eqid 2610 | . . 3 ⊢ (algSc‘𝐾) = (algSc‘𝐾) | |
17 | asclpropd.f | . . 3 ⊢ 𝐹 = (Scalar‘𝐾) | |
18 | eqid 2610 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
19 | eqid 2610 | . . 3 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾) | |
20 | eqid 2610 | . . 3 ⊢ (1r‘𝐾) = (1r‘𝐾) | |
21 | 16, 17, 18, 19, 20 | asclfval 19155 | . 2 ⊢ (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) |
22 | eqid 2610 | . . 3 ⊢ (algSc‘𝐿) = (algSc‘𝐿) | |
23 | asclpropd.g | . . 3 ⊢ 𝐺 = (Scalar‘𝐿) | |
24 | eqid 2610 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
25 | eqid 2610 | . . 3 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿) | |
26 | eqid 2610 | . . 3 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
27 | 22, 23, 24, 25, 26 | asclfval 19155 | . 2 ⊢ (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
28 | 15, 21, 27 | 3eqtr4g 2669 | 1 ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 1rcur 18324 algSccascl 19132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-slot 15699 df-base 15700 df-ascl 19135 |
This theorem is referenced by: ply1ascl 19449 |
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