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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldgenval | Structured version Visualization version GIF version |
Description: Value of the field generating function: (πΉ fldGen π) is the smallest sub-division-ring of πΉ containing π. (Contributed by Thierry Arnoux, 11-Jan-2025.) |
Ref | Expression |
---|---|
fldgenval.1 | β’ π΅ = (BaseβπΉ) |
fldgenval.2 | β’ (π β πΉ β DivRing) |
fldgenval.3 | β’ (π β π β π΅) |
Ref | Expression |
---|---|
fldgenval | β’ (π β (πΉ fldGen π) = β© {π β (SubDRingβπΉ) β£ π β π}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fldgenval.2 | . . 3 β’ (π β πΉ β DivRing) | |
2 | 1 | elexd 3467 | . 2 β’ (π β πΉ β V) |
3 | fldgenval.1 | . . . . 5 β’ π΅ = (BaseβπΉ) | |
4 | 3 | fvexi 6860 | . . . 4 β’ π΅ β V |
5 | 4 | a1i 11 | . . 3 β’ (π β π΅ β V) |
6 | fldgenval.3 | . . 3 β’ (π β π β π΅) | |
7 | 5, 6 | ssexd 5285 | . 2 β’ (π β π β V) |
8 | 3 | sdrgid 20305 | . . . . 5 β’ (πΉ β DivRing β π΅ β (SubDRingβπΉ)) |
9 | 1, 8 | syl 17 | . . . 4 β’ (π β π΅ β (SubDRingβπΉ)) |
10 | sseq2 3974 | . . . . 5 β’ (π = π΅ β (π β π β π β π΅)) | |
11 | 10 | adantl 483 | . . . 4 β’ ((π β§ π = π΅) β (π β π β π β π΅)) |
12 | 9, 11, 6 | rspcedvd 3585 | . . 3 β’ (π β βπ β (SubDRingβπΉ)π β π) |
13 | intexrab 5301 | . . 3 β’ (βπ β (SubDRingβπΉ)π β π β β© {π β (SubDRingβπΉ) β£ π β π} β V) | |
14 | 12, 13 | sylib 217 | . 2 β’ (π β β© {π β (SubDRingβπΉ) β£ π β π} β V) |
15 | simpl 484 | . . . . . 6 β’ ((π = πΉ β§ π = π) β π = πΉ) | |
16 | 15 | fveq2d 6850 | . . . . 5 β’ ((π = πΉ β§ π = π) β (SubDRingβπ) = (SubDRingβπΉ)) |
17 | simpr 486 | . . . . . 6 β’ ((π = πΉ β§ π = π) β π = π) | |
18 | 17 | sseq1d 3979 | . . . . 5 β’ ((π = πΉ β§ π = π) β (π β π β π β π)) |
19 | 16, 18 | rabeqbidv 3423 | . . . 4 β’ ((π = πΉ β§ π = π) β {π β (SubDRingβπ) β£ π β π} = {π β (SubDRingβπΉ) β£ π β π}) |
20 | 19 | inteqd 4916 | . . 3 β’ ((π = πΉ β§ π = π) β β© {π β (SubDRingβπ) β£ π β π} = β© {π β (SubDRingβπΉ) β£ π β π}) |
21 | df-fldgen 32134 | . . 3 β’ fldGen = (π β V, π β V β¦ β© {π β (SubDRingβπ) β£ π β π}) | |
22 | 20, 21 | ovmpoga 7513 | . 2 β’ ((πΉ β V β§ π β V β§ β© {π β (SubDRingβπΉ) β£ π β π} β V) β (πΉ fldGen π) = β© {π β (SubDRingβπΉ) β£ π β π}) |
23 | 2, 7, 14, 22 | syl3anc 1372 | 1 β’ (π β (πΉ fldGen π) = β© {π β (SubDRingβπΉ) β£ π β π}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3070 {crab 3406 Vcvv 3447 β wss 3914 β© cint 4911 βcfv 6500 (class class class)co 7361 Basecbs 17091 DivRingcdr 20219 SubDRingcsdrg 20302 fldGen cfldgen 32133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mgp 19905 df-ur 19922 df-ring 19974 df-drng 20221 df-subrg 20262 df-sdrg 20303 df-fldgen 32134 |
This theorem is referenced by: fldgenssid 32136 fldgensdrg 32137 fldgenss 32138 fldgenidfld 32139 fldgenid 32140 primefldgen1 32142 |
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