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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 21200 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(ringLMod‘𝐾)) | |
| 3 | 1, 2 | eqtrdi 2793 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ringLMod‘𝐾))) |
| 4 | lidlpropd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
| 5 | rlmbas 21200 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(ringLMod‘𝐿)) | |
| 6 | 4, 5 | eqtrdi 2793 | . . . 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 21201 | . . . . . 6 ⊢ (+g‘𝐾) = (+g‘(ringLMod‘𝐾)) | |
| 10 | 9 | oveqi 7444 | . . . . 5 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(ringLMod‘𝐾))𝑦) |
| 11 | rlmplusg 21201 | . . . . . 6 ⊢ (+g‘𝐿) = (+g‘(ringLMod‘𝐿)) | |
| 12 | 11 | oveqi 7444 | . . . . 5 ⊢ (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦) |
| 13 | 8, 10, 12 | 3eqtr3g 2800 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘(ringLMod‘𝐾))𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦)) |
| 14 | rlmvsca 21207 | . . . . . 6 ⊢ (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾)) | |
| 15 | 14 | oveqi 7444 | . . . . 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 2846 | . . . 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 21207 | . . . . . 6 ⊢ (.r‘𝐿) = ( ·𝑠 ‘(ringLMod‘𝐿)) | |
| 20 | 19 | oveqi 7444 | . . . . 5 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦) |
| 21 | 18, 15, 20 | 3eqtr3g 2800 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦)) |
| 22 | baseid 17250 | . . . . . . 7 ⊢ Base = Slot (Base‘ndx) | |
| 23 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 24 | 22, 23 | strfvi 17227 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘( I ‘𝐾)) |
| 25 | rlmsca2 21206 | . . . . . . 7 ⊢ ( I ‘𝐾) = (Scalar‘(ringLMod‘𝐾)) | |
| 26 | 25 | fveq2i 6909 | . . . . . 6 ⊢ (Base‘( I ‘𝐾)) = (Base‘(Scalar‘(ringLMod‘𝐾))) |
| 27 | 24, 26 | eqtri 2765 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(Scalar‘(ringLMod‘𝐾))) |
| 28 | 1, 27 | eqtrdi 2793 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐾)))) |
| 29 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
| 30 | 22, 29 | strfvi 17227 | . . . . . 6 ⊢ (Base‘𝐿) = (Base‘( I ‘𝐿)) |
| 31 | rlmsca2 21206 | . . . . . . 7 ⊢ ( I ‘𝐿) = (Scalar‘(ringLMod‘𝐿)) | |
| 32 | 31 | fveq2i 6909 | . . . . . 6 ⊢ (Base‘( I ‘𝐿)) = (Base‘(Scalar‘(ringLMod‘𝐿))) |
| 33 | 30, 32 | eqtri 2765 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(Scalar‘(ringLMod‘𝐿))) |
| 34 | 4, 33 | eqtrdi 2793 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐿)))) |
| 35 | 3, 6, 7, 13, 17, 21, 28, 34 | lsspropd 21016 | . . 3 ⊢ (𝜑 → (LSubSp‘(ringLMod‘𝐾)) = (LSubSp‘(ringLMod‘𝐿))) |
| 36 | lidlval 21220 | . . 3 ⊢ (LIdeal‘𝐾) = (LSubSp‘(ringLMod‘𝐾)) | |
| 37 | lidlval 21220 | . . 3 ⊢ (LIdeal‘𝐿) = (LSubSp‘(ringLMod‘𝐿)) | |
| 38 | 35, 36, 37 | 3eqtr4g 2802 | . 2 ⊢ (𝜑 → (LIdeal‘𝐾) = (LIdeal‘𝐿)) |
| 39 | fvexd 6921 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝐾) ∈ V) | |
| 40 | fvexd 6921 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝐿) ∈ V) | |
| 41 | 3, 6, 7, 13, 17, 21, 28, 34, 39, 40 | lsppropd 21017 | . . 3 ⊢ (𝜑 → (LSpan‘(ringLMod‘𝐾)) = (LSpan‘(ringLMod‘𝐿))) |
| 42 | rspval 21221 | . . 3 ⊢ (RSpan‘𝐾) = (LSpan‘(ringLMod‘𝐾)) | |
| 43 | rspval 21221 | . . 3 ⊢ (RSpan‘𝐿) = (LSpan‘(ringLMod‘𝐿)) | |
| 44 | 41, 42, 43 | 3eqtr4g 2802 | . 2 ⊢ (𝜑 → (RSpan‘𝐾) = (RSpan‘𝐿)) |
| 45 | 38, 44 | jca 511 | 1 ⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 I cid 5577 ‘cfv 6561 (class class class)co 7431 ndxcnx 17230 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 LSubSpclss 20929 LSpanclspn 20969 ringLModcrglmod 21171 LIdealclidl 21216 RSpancrsp 21217 |
| 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-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| 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-nel 3047 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-int 4947 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-sca 17313 df-vsca 17314 df-ip 17315 df-lss 20930 df-lsp 20970 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 |
| This theorem is referenced by: crngridl 21290 |
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