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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspsnel | Structured version Visualization version GIF version |
Description: Membership in a principal ideal. Analogous to lspsnel 19775. (Contributed by Thierry Arnoux, 15-Jan-2024.) |
Ref | Expression |
---|---|
rspsnel.1 | ⊢ 𝐵 = (Base‘𝑅) |
rspsnel.2 | ⊢ · = (.r‘𝑅) |
rspsnel.3 | ⊢ 𝐾 = (RSpan‘𝑅) |
Ref | Expression |
---|---|
rspsnel | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥 ∈ 𝐵 𝐼 = (𝑥 · 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmlmod 19977 | . . 3 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
2 | simpr 487 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
3 | rspsnel.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | eleqtrdi 2923 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (Base‘𝑅)) |
5 | eqid 2821 | . . . 4 ⊢ (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅)) | |
6 | eqid 2821 | . . . 4 ⊢ (Base‘(Scalar‘(ringLMod‘𝑅))) = (Base‘(Scalar‘(ringLMod‘𝑅))) | |
7 | rlmbas 19967 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
8 | rspsnel.2 | . . . . 5 ⊢ · = (.r‘𝑅) | |
9 | rlmvsca 19974 | . . . . 5 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
10 | 8, 9 | eqtri 2844 | . . . 4 ⊢ · = ( ·𝑠 ‘(ringLMod‘𝑅)) |
11 | rspsnel.3 | . . . . 5 ⊢ 𝐾 = (RSpan‘𝑅) | |
12 | rspval 19965 | . . . . 5 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
13 | 11, 12 | eqtri 2844 | . . . 4 ⊢ 𝐾 = (LSpan‘(ringLMod‘𝑅)) |
14 | 5, 6, 7, 10, 13 | lspsnel 19775 | . . 3 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝑋 ∈ (Base‘𝑅)) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))𝐼 = (𝑥 · 𝑋))) |
15 | 1, 4, 14 | syl2an2r 683 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))𝐼 = (𝑥 · 𝑋))) |
16 | rlmsca 19972 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | |
17 | 16 | adantr 483 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 = (Scalar‘(ringLMod‘𝑅))) |
18 | 17 | fveq2d 6674 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (Base‘𝑅) = (Base‘(Scalar‘(ringLMod‘𝑅)))) |
19 | 3, 18 | syl5req 2869 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (Base‘(Scalar‘(ringLMod‘𝑅))) = 𝐵) |
20 | 19 | rexeqdv 3416 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (∃𝑥 ∈ (Base‘(Scalar‘(ringLMod‘𝑅)))𝐼 = (𝑥 · 𝑋) ↔ ∃𝑥 ∈ 𝐵 𝐼 = (𝑥 · 𝑋))) |
21 | 15, 20 | bitrd 281 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥 ∈ 𝐵 𝐼 = (𝑥 · 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 {csn 4567 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 .rcmulr 16566 Scalarcsca 16568 ·𝑠 cvsca 16569 Ringcrg 19297 LModclmod 19634 LSpanclspn 19743 ringLModcrglmod 19941 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-rmo 3146 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-1st 7689 df-2nd 7690 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-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-mgp 19240 df-ur 19252 df-ring 19299 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-sra 19944 df-rgmod 19945 df-rsp 19947 |
This theorem is referenced by: lsmsnpridl 30948 isprmidlc 30963 |
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