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Mirrors > Home > MPE Home > Th. List > lidlrsppropd | Structured version Visualization version GIF version |
Description: The left ideals and ring span of a ring depend only on the ring components. Here 𝑊 is expected to be either 𝐵 (when closure is available) or V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
lidlpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
lidlpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
lidlpropd.3 | ⊢ (𝜑 → 𝐵 ⊆ 𝑊) |
lidlpropd.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
lidlpropd.5 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) ∈ 𝑊) |
lidlpropd.6 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
Ref | Expression |
---|---|
lidlrsppropd | ⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidlpropd.1 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | rlmbas 19960 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(ringLMod‘𝐾)) | |
3 | 1, 2 | eqtrdi 2849 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ringLMod‘𝐾))) |
4 | lidlpropd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
5 | rlmbas 19960 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(ringLMod‘𝐿)) | |
6 | 4, 5 | eqtrdi 2849 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ringLMod‘𝐿))) |
7 | lidlpropd.3 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑊) | |
8 | lidlpropd.4 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
9 | rlmplusg 19961 | . . . . . 6 ⊢ (+g‘𝐾) = (+g‘(ringLMod‘𝐾)) | |
10 | 9 | oveqi 7148 | . . . . 5 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(ringLMod‘𝐾))𝑦) |
11 | rlmplusg 19961 | . . . . . 6 ⊢ (+g‘𝐿) = (+g‘(ringLMod‘𝐿)) | |
12 | 11 | oveqi 7148 | . . . . 5 ⊢ (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦) |
13 | 8, 10, 12 | 3eqtr3g 2856 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘(ringLMod‘𝐾))𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦)) |
14 | rlmvsca 19967 | . . . . . 6 ⊢ (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾)) | |
15 | 14 | oveqi 7148 | . . . . 5 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) |
16 | lidlpropd.5 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) ∈ 𝑊) | |
17 | 15, 16 | eqeltrrid 2895 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) ∈ 𝑊) |
18 | lidlpropd.6 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) | |
19 | rlmvsca 19967 | . . . . . 6 ⊢ (.r‘𝐿) = ( ·𝑠 ‘(ringLMod‘𝐿)) | |
20 | 19 | oveqi 7148 | . . . . 5 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦) |
21 | 18, 15, 20 | 3eqtr3g 2856 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦)) |
22 | baseid 16535 | . . . . . . 7 ⊢ Base = Slot (Base‘ndx) | |
23 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
24 | 22, 23 | strfvi 16529 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘( I ‘𝐾)) |
25 | rlmsca2 19966 | . . . . . . 7 ⊢ ( I ‘𝐾) = (Scalar‘(ringLMod‘𝐾)) | |
26 | 25 | fveq2i 6648 | . . . . . 6 ⊢ (Base‘( I ‘𝐾)) = (Base‘(Scalar‘(ringLMod‘𝐾))) |
27 | 24, 26 | eqtri 2821 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(Scalar‘(ringLMod‘𝐾))) |
28 | 1, 27 | eqtrdi 2849 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐾)))) |
29 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
30 | 22, 29 | strfvi 16529 | . . . . . 6 ⊢ (Base‘𝐿) = (Base‘( I ‘𝐿)) |
31 | rlmsca2 19966 | . . . . . . 7 ⊢ ( I ‘𝐿) = (Scalar‘(ringLMod‘𝐿)) | |
32 | 31 | fveq2i 6648 | . . . . . 6 ⊢ (Base‘( I ‘𝐿)) = (Base‘(Scalar‘(ringLMod‘𝐿))) |
33 | 30, 32 | eqtri 2821 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(Scalar‘(ringLMod‘𝐿))) |
34 | 4, 33 | eqtrdi 2849 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐿)))) |
35 | 3, 6, 7, 13, 17, 21, 28, 34 | lsspropd 19782 | . . 3 ⊢ (𝜑 → (LSubSp‘(ringLMod‘𝐾)) = (LSubSp‘(ringLMod‘𝐿))) |
36 | lidlval 19957 | . . 3 ⊢ (LIdeal‘𝐾) = (LSubSp‘(ringLMod‘𝐾)) | |
37 | lidlval 19957 | . . 3 ⊢ (LIdeal‘𝐿) = (LSubSp‘(ringLMod‘𝐿)) | |
38 | 35, 36, 37 | 3eqtr4g 2858 | . 2 ⊢ (𝜑 → (LIdeal‘𝐾) = (LIdeal‘𝐿)) |
39 | fvexd 6660 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝐾) ∈ V) | |
40 | fvexd 6660 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝐿) ∈ V) | |
41 | 3, 6, 7, 13, 17, 21, 28, 34, 39, 40 | lsppropd 19783 | . . 3 ⊢ (𝜑 → (LSpan‘(ringLMod‘𝐾)) = (LSpan‘(ringLMod‘𝐿))) |
42 | rspval 19958 | . . 3 ⊢ (RSpan‘𝐾) = (LSpan‘(ringLMod‘𝐾)) | |
43 | rspval 19958 | . . 3 ⊢ (RSpan‘𝐿) = (LSpan‘(ringLMod‘𝐿)) | |
44 | 41, 42, 43 | 3eqtr4g 2858 | . 2 ⊢ (𝜑 → (RSpan‘𝐾) = (RSpan‘𝐿)) |
45 | 38, 44 | jca 515 | 1 ⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 I cid 5424 ‘cfv 6324 (class class class)co 7135 ndxcnx 16472 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 LSubSpclss 19696 LSpanclspn 19736 ringLModcrglmod 19934 LIdealclidl 19935 RSpancrsp 19936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 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-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-sca 16573 df-vsca 16574 df-ip 16575 df-lss 19697 df-lsp 19737 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-rsp 19940 |
This theorem is referenced by: crngridl 20004 |
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