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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldgenss | Structured version Visualization version GIF version |
Description: Generated subfields preserve subset ordering. ( see lspss 20816 and spanss 31025) (Contributed by Thierry Arnoux, 15-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΅) |
fldgenss.t | β’ (π β π β π) |
Ref | Expression |
---|---|
fldgenss | β’ (π β (πΉ fldGen π) β (πΉ fldGen π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fldgenss.t | . . . . . . . 8 β’ (π β π β π) | |
2 | 1 | adantr 480 | . . . . . . 7 β’ ((π β§ π β π) β π β π) |
3 | simpr 484 | . . . . . . 7 β’ ((π β§ π β π) β π β π) | |
4 | 2, 3 | sstrd 3984 | . . . . . 6 β’ ((π β§ π β π) β π β π) |
5 | 4 | ex 412 | . . . . 5 β’ (π β (π β π β π β π)) |
6 | 5 | adantr 480 | . . . 4 β’ ((π β§ π β (SubDRingβπΉ)) β (π β π β π β π)) |
7 | 6 | ss2rabdv 4065 | . . 3 β’ (π β {π β (SubDRingβπΉ) β£ π β π} β {π β (SubDRingβπΉ) β£ π β π}) |
8 | intss 4963 | . . 3 β’ ({π β (SubDRingβπΉ) β£ π β π} β {π β (SubDRingβπΉ) β£ π β π} β β© {π β (SubDRingβπΉ) β£ π β π} β β© {π β (SubDRingβπΉ) β£ π β π}) | |
9 | 7, 8 | syl 17 | . 2 β’ (π β β© {π β (SubDRingβπΉ) β£ π β π} β β© {π β (SubDRingβπΉ) β£ π β π}) |
10 | fldgenval.1 | . . 3 β’ π΅ = (BaseβπΉ) | |
11 | fldgenval.2 | . . 3 β’ (π β πΉ β DivRing) | |
12 | fldgenval.3 | . . . 4 β’ (π β π β π΅) | |
13 | 1, 12 | sstrd 3984 | . . 3 β’ (π β π β π΅) |
14 | 10, 11, 13 | fldgenval 32829 | . 2 β’ (π β (πΉ fldGen π) = β© {π β (SubDRingβπΉ) β£ π β π}) |
15 | 10, 11, 12 | fldgenval 32829 | . 2 β’ (π β (πΉ fldGen π) = β© {π β (SubDRingβπΉ) β£ π β π}) |
16 | 9, 14, 15 | 3sstr4d 4021 | 1 β’ (π β (πΉ fldGen π) β (πΉ fldGen π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3424 β wss 3940 β© cint 4940 βcfv 6533 (class class class)co 7401 Basecbs 17140 DivRingcdr 20572 SubDRingcsdrg 20622 fldGen cfldgen 32827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-0g 17383 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-mgp 20025 df-ur 20072 df-ring 20125 df-subrg 20456 df-drng 20574 df-sdrg 20623 df-fldgen 32828 |
This theorem is referenced by: 1fldgenq 32839 |
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