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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 19967 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(ringLMod‘𝐾)) | |
3 | 1, 2 | syl6eq 2872 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ringLMod‘𝐾))) |
4 | lidlpropd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
5 | rlmbas 19967 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(ringLMod‘𝐿)) | |
6 | 4, 5 | syl6eq 2872 | . . . 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 19968 | . . . . . 6 ⊢ (+g‘𝐾) = (+g‘(ringLMod‘𝐾)) | |
10 | 9 | oveqi 7169 | . . . . 5 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(ringLMod‘𝐾))𝑦) |
11 | rlmplusg 19968 | . . . . . 6 ⊢ (+g‘𝐿) = (+g‘(ringLMod‘𝐿)) | |
12 | 11 | oveqi 7169 | . . . . 5 ⊢ (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦) |
13 | 8, 10, 12 | 3eqtr3g 2879 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘(ringLMod‘𝐾))𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦)) |
14 | rlmvsca 19974 | . . . . . 6 ⊢ (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾)) | |
15 | 14 | oveqi 7169 | . . . . 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 2918 | . . . 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 19974 | . . . . . 6 ⊢ (.r‘𝐿) = ( ·𝑠 ‘(ringLMod‘𝐿)) | |
20 | 19 | oveqi 7169 | . . . . 5 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦) |
21 | 18, 15, 20 | 3eqtr3g 2879 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦)) |
22 | baseid 16543 | . . . . . . 7 ⊢ Base = Slot (Base‘ndx) | |
23 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
24 | 22, 23 | strfvi 16537 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘( I ‘𝐾)) |
25 | rlmsca2 19973 | . . . . . . 7 ⊢ ( I ‘𝐾) = (Scalar‘(ringLMod‘𝐾)) | |
26 | 25 | fveq2i 6673 | . . . . . 6 ⊢ (Base‘( I ‘𝐾)) = (Base‘(Scalar‘(ringLMod‘𝐾))) |
27 | 24, 26 | eqtri 2844 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(Scalar‘(ringLMod‘𝐾))) |
28 | 1, 27 | syl6eq 2872 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐾)))) |
29 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
30 | 22, 29 | strfvi 16537 | . . . . . 6 ⊢ (Base‘𝐿) = (Base‘( I ‘𝐿)) |
31 | rlmsca2 19973 | . . . . . . 7 ⊢ ( I ‘𝐿) = (Scalar‘(ringLMod‘𝐿)) | |
32 | 31 | fveq2i 6673 | . . . . . 6 ⊢ (Base‘( I ‘𝐿)) = (Base‘(Scalar‘(ringLMod‘𝐿))) |
33 | 30, 32 | eqtri 2844 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(Scalar‘(ringLMod‘𝐿))) |
34 | 4, 33 | syl6eq 2872 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐿)))) |
35 | 3, 6, 7, 13, 17, 21, 28, 34 | lsspropd 19789 | . . 3 ⊢ (𝜑 → (LSubSp‘(ringLMod‘𝐾)) = (LSubSp‘(ringLMod‘𝐿))) |
36 | lidlval 19964 | . . 3 ⊢ (LIdeal‘𝐾) = (LSubSp‘(ringLMod‘𝐾)) | |
37 | lidlval 19964 | . . 3 ⊢ (LIdeal‘𝐿) = (LSubSp‘(ringLMod‘𝐿)) | |
38 | 35, 36, 37 | 3eqtr4g 2881 | . 2 ⊢ (𝜑 → (LIdeal‘𝐾) = (LIdeal‘𝐿)) |
39 | fvexd 6685 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝐾) ∈ V) | |
40 | fvexd 6685 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝐿) ∈ V) | |
41 | 3, 6, 7, 13, 17, 21, 28, 34, 39, 40 | lsppropd 19790 | . . 3 ⊢ (𝜑 → (LSpan‘(ringLMod‘𝐾)) = (LSpan‘(ringLMod‘𝐿))) |
42 | rspval 19965 | . . 3 ⊢ (RSpan‘𝐾) = (LSpan‘(ringLMod‘𝐾)) | |
43 | rspval 19965 | . . 3 ⊢ (RSpan‘𝐿) = (LSpan‘(ringLMod‘𝐿)) | |
44 | 41, 42, 43 | 3eqtr4g 2881 | . 2 ⊢ (𝜑 → (RSpan‘𝐾) = (RSpan‘𝐿)) |
45 | 38, 44 | jca 514 | 1 ⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 I cid 5459 ‘cfv 6355 (class class class)co 7156 ndxcnx 16480 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 Scalarcsca 16568 ·𝑠 cvsca 16569 LSubSpclss 19703 LSpanclspn 19743 ringLModcrglmod 19941 LIdealclidl 19942 RSpancrsp 19943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-sca 16581 df-vsca 16582 df-ip 16583 df-lss 19704 df-lsp 19744 df-sra 19944 df-rgmod 19945 df-lidl 19946 df-rsp 19947 |
This theorem is referenced by: crngridl 20011 |
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