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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 7458 | . . . . . . 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 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 7447 | . . . . . 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 2777 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
| 10 | 9 | mpteq2dva 5242 | . . 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 5237 | . . 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 5237 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))) |
| 15 | 10, 12, 14 | 3eqtr3d 2785 | . 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 21899 | . 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 21899 | . 2 ⊢ (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
| 28 | 15, 21, 27 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 1rcur 20178 algSccascl 21872 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-ascl 21875 |
| This theorem is referenced by: ply1ascl 22261 |
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