Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7172 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑃 ∧ (1r‘𝐾) ∈ 𝑊)) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾))) |
4 | 3 | anassrs 468 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑃) ∧ (1r‘𝐾) ∈ 𝑊) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾))) |
5 | 1, 4 | mpidan 685 | . . . . 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 7161 | . . . . . 6 ⊢ (𝜑 → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
8 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐿)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
9 | 5, 8 | eqtrd 2853 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑃) → (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾)) = (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
10 | 9 | mpteq2dva 5152 | . . 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 5146 | . . 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 5146 | . . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑃 ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))) |
15 | 10, 12, 14 | 3eqtr3d 2861 | . 2 ⊢ (𝜑 → (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿)))) |
16 | eqid 2818 | . . 3 ⊢ (algSc‘𝐾) = (algSc‘𝐾) | |
17 | asclpropd.f | . . 3 ⊢ 𝐹 = (Scalar‘𝐾) | |
18 | eqid 2818 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
19 | eqid 2818 | . . 3 ⊢ ( ·𝑠 ‘𝐾) = ( ·𝑠 ‘𝐾) | |
20 | eqid 2818 | . . 3 ⊢ (1r‘𝐾) = (1r‘𝐾) | |
21 | 16, 17, 18, 19, 20 | asclfval 20036 | . 2 ⊢ (algSc‘𝐾) = (𝑧 ∈ (Base‘𝐹) ↦ (𝑧( ·𝑠 ‘𝐾)(1r‘𝐾))) |
22 | eqid 2818 | . . 3 ⊢ (algSc‘𝐿) = (algSc‘𝐿) | |
23 | asclpropd.g | . . 3 ⊢ 𝐺 = (Scalar‘𝐿) | |
24 | eqid 2818 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
25 | eqid 2818 | . . 3 ⊢ ( ·𝑠 ‘𝐿) = ( ·𝑠 ‘𝐿) | |
26 | eqid 2818 | . . 3 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
27 | 22, 23, 24, 25, 26 | asclfval 20036 | . 2 ⊢ (algSc‘𝐿) = (𝑧 ∈ (Base‘𝐺) ↦ (𝑧( ·𝑠 ‘𝐿)(1r‘𝐿))) |
28 | 15, 21, 27 | 3eqtr4g 2878 | 1 ⊢ (𝜑 → (algSc‘𝐾) = (algSc‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 1rcur 19180 algSccascl 20012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-slot 16475 df-base 16477 df-ascl 20015 |
This theorem is referenced by: ply1ascl 20354 |
Copyright terms: Public domain | W3C validator |