Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idlsrgbas | Structured version Visualization version GIF version |
Description: Baae of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
Ref | Expression |
---|---|
idlsrgbas.1 | ⊢ 𝑆 = (IDLsrg‘𝑅) |
idlsrgbas.2 | ⊢ 𝐼 = (LIdeal‘𝑅) |
Ref | Expression |
---|---|
idlsrgbas | ⊢ (𝑅 ∈ 𝑉 → 𝐼 = (Base‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlsrgbas.2 | . . . 4 ⊢ 𝐼 = (LIdeal‘𝑅) | |
2 | 1 | fvexi 6709 | . . 3 ⊢ 𝐼 ∈ V |
3 | eqid 2736 | . . . . 5 ⊢ ({〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}〉}) = ({〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}〉}) | |
4 | 3 | idlsrgstr 31315 | . . . 4 ⊢ ({〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}〉}) Struct 〈1, ;10〉 |
5 | baseid 16724 | . . . 4 ⊢ Base = Slot (Base‘ndx) | |
6 | snsstp1 4715 | . . . . 5 ⊢ {〈(Base‘ndx), 𝐼〉} ⊆ {〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} | |
7 | ssun1 4072 | . . . . 5 ⊢ {〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} ⊆ ({〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}〉}) | |
8 | 6, 7 | sstri 3896 | . . . 4 ⊢ {〈(Base‘ndx), 𝐼〉} ⊆ ({〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}〉}) |
9 | 4, 5, 8 | strfv 16713 | . . 3 ⊢ (𝐼 ∈ V → 𝐼 = (Base‘({〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}〉}))) |
10 | 2, 9 | ax-mp 5 | . 2 ⊢ 𝐼 = (Base‘({〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}〉})) |
11 | idlsrgbas.1 | . . . 4 ⊢ 𝑆 = (IDLsrg‘𝑅) | |
12 | eqid 2736 | . . . . 5 ⊢ (LSSum‘𝑅) = (LSSum‘𝑅) | |
13 | eqid 2736 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
14 | eqid 2736 | . . . . 5 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅)) | |
15 | 1, 12, 13, 14 | idlsrgval 31316 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (IDLsrg‘𝑅) = ({〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}〉})) |
16 | 11, 15 | syl5eq 2783 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = ({〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}〉})) |
17 | 16 | fveq2d 6699 | . 2 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑆) = (Base‘({〈(Base‘ndx), 𝐼〉, 〈(+g‘ndx), (LSSum‘𝑅)〉, 〈(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))〉} ∪ {〈(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})〉, 〈(le‘ndx), {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}〉}))) |
18 | 10, 17 | eqtr4id 2790 | 1 ⊢ (𝑅 ∈ 𝑉 → 𝐼 = (Base‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {crab 3055 Vcvv 3398 ∪ cun 3851 ⊆ wss 3853 {csn 4527 {cpr 4529 {ctp 4531 〈cop 4533 {copab 5101 ↦ cmpt 5120 ran crn 5537 ‘cfv 6358 (class class class)co 7191 ∈ cmpo 7193 0cc0 10694 1c1 10695 ;cdc 12258 ndxcnx 16663 Basecbs 16666 +gcplusg 16749 .rcmulr 16750 TopSetcts 16755 lecple 16756 LSSumclsm 18977 mulGrpcmgp 19458 LIdealclidl 20161 RSpancrsp 20162 IDLsrgcidlsrg 31313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-plusg 16762 df-mulr 16763 df-tset 16768 df-ple 16769 df-idlsrg 31314 |
This theorem is referenced by: idlsrg0g 31319 idlsrgmnd 31327 idlsrgcmnd 31328 rspecbas 31483 rspectopn 31485 |
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