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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 21085 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(ringLMod‘𝐾)) | |
3 | 1, 2 | eqtrdi 2781 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ringLMod‘𝐾))) |
4 | lidlpropd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
5 | rlmbas 21085 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(ringLMod‘𝐿)) | |
6 | 4, 5 | eqtrdi 2781 | . . . 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 21086 | . . . . . 6 ⊢ (+g‘𝐾) = (+g‘(ringLMod‘𝐾)) | |
10 | 9 | oveqi 7426 | . . . . 5 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(ringLMod‘𝐾))𝑦) |
11 | rlmplusg 21086 | . . . . . 6 ⊢ (+g‘𝐿) = (+g‘(ringLMod‘𝐿)) | |
12 | 11 | oveqi 7426 | . . . . 5 ⊢ (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦) |
13 | 8, 10, 12 | 3eqtr3g 2788 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘(ringLMod‘𝐾))𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦)) |
14 | rlmvsca 21092 | . . . . . 6 ⊢ (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾)) | |
15 | 14 | oveqi 7426 | . . . . 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 2830 | . . . 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 21092 | . . . . . 6 ⊢ (.r‘𝐿) = ( ·𝑠 ‘(ringLMod‘𝐿)) | |
20 | 19 | oveqi 7426 | . . . . 5 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦) |
21 | 18, 15, 20 | 3eqtr3g 2788 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦)) |
22 | baseid 17177 | . . . . . . 7 ⊢ Base = Slot (Base‘ndx) | |
23 | eqid 2725 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
24 | 22, 23 | strfvi 17153 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘( I ‘𝐾)) |
25 | rlmsca2 21091 | . . . . . . 7 ⊢ ( I ‘𝐾) = (Scalar‘(ringLMod‘𝐾)) | |
26 | 25 | fveq2i 6893 | . . . . . 6 ⊢ (Base‘( I ‘𝐾)) = (Base‘(Scalar‘(ringLMod‘𝐾))) |
27 | 24, 26 | eqtri 2753 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(Scalar‘(ringLMod‘𝐾))) |
28 | 1, 27 | eqtrdi 2781 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐾)))) |
29 | eqid 2725 | . . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
30 | 22, 29 | strfvi 17153 | . . . . . 6 ⊢ (Base‘𝐿) = (Base‘( I ‘𝐿)) |
31 | rlmsca2 21091 | . . . . . . 7 ⊢ ( I ‘𝐿) = (Scalar‘(ringLMod‘𝐿)) | |
32 | 31 | fveq2i 6893 | . . . . . 6 ⊢ (Base‘( I ‘𝐿)) = (Base‘(Scalar‘(ringLMod‘𝐿))) |
33 | 30, 32 | eqtri 2753 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(Scalar‘(ringLMod‘𝐿))) |
34 | 4, 33 | eqtrdi 2781 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐿)))) |
35 | 3, 6, 7, 13, 17, 21, 28, 34 | lsspropd 20901 | . . 3 ⊢ (𝜑 → (LSubSp‘(ringLMod‘𝐾)) = (LSubSp‘(ringLMod‘𝐿))) |
36 | lidlval 21105 | . . 3 ⊢ (LIdeal‘𝐾) = (LSubSp‘(ringLMod‘𝐾)) | |
37 | lidlval 21105 | . . 3 ⊢ (LIdeal‘𝐿) = (LSubSp‘(ringLMod‘𝐿)) | |
38 | 35, 36, 37 | 3eqtr4g 2790 | . 2 ⊢ (𝜑 → (LIdeal‘𝐾) = (LIdeal‘𝐿)) |
39 | fvexd 6905 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝐾) ∈ V) | |
40 | fvexd 6905 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝐿) ∈ V) | |
41 | 3, 6, 7, 13, 17, 21, 28, 34, 39, 40 | lsppropd 20902 | . . 3 ⊢ (𝜑 → (LSpan‘(ringLMod‘𝐾)) = (LSpan‘(ringLMod‘𝐿))) |
42 | rspval 21106 | . . 3 ⊢ (RSpan‘𝐾) = (LSpan‘(ringLMod‘𝐾)) | |
43 | rspval 21106 | . . 3 ⊢ (RSpan‘𝐿) = (LSpan‘(ringLMod‘𝐿)) | |
44 | 41, 42, 43 | 3eqtr4g 2790 | . 2 ⊢ (𝜑 → (RSpan‘𝐾) = (RSpan‘𝐿)) |
45 | 38, 44 | jca 510 | 1 ⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3941 I cid 5570 ‘cfv 6543 (class class class)co 7413 ndxcnx 17156 Basecbs 17174 +gcplusg 17227 .rcmulr 17228 Scalarcsca 17230 ·𝑠 cvsca 17231 LSubSpclss 20814 LSpanclspn 20854 ringLModcrglmod 21056 LIdealclidl 21101 RSpancrsp 21102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-sca 17243 df-vsca 17244 df-ip 17245 df-lss 20815 df-lsp 20855 df-sra 21057 df-rgmod 21058 df-lidl 21103 df-rsp 21104 |
This theorem is referenced by: crngridl 21171 |
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