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Mirrors > Home > ILE Home > Th. List > srabaseg | GIF version |
Description: Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
Ref | Expression |
---|---|
srapart.a | β’ (π β π΄ = ((subringAlg βπ)βπ)) |
srapart.s | β’ (π β π β (Baseβπ)) |
srapart.ex | β’ (π β π β π) |
Ref | Expression |
---|---|
srabaseg | β’ (π β (Baseβπ) = (Baseβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srapart.a | . 2 β’ (π β π΄ = ((subringAlg βπ)βπ)) | |
2 | srapart.s | . 2 β’ (π β π β (Baseβπ)) | |
3 | srapart.ex | . 2 β’ (π β π β π) | |
4 | baseslid 12544 | . 2 β’ (Base = Slot (Baseβndx) β§ (Baseβndx) β β) | |
5 | scandxnbasendx 12638 | . 2 β’ (Scalarβndx) β (Baseβndx) | |
6 | vscandxnbasendx 12643 | . 2 β’ ( Β·π βndx) β (Baseβndx) | |
7 | ipndxnbasendx 12656 | . 2 β’ (Β·πβndx) β (Baseβndx) | |
8 | 1, 2, 3, 4, 5, 6, 7 | sralemg 13722 | 1 β’ (π β (Baseβπ) = (Baseβπ΄)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1364 β wcel 2160 β wss 3144 βcfv 5232 Basecbs 12487 subringAlg csra 13717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7922 ax-resscn 7923 ax-1cn 7924 ax-1re 7925 ax-icn 7926 ax-addcl 7927 ax-addrcl 7928 ax-mulcl 7929 ax-addcom 7931 ax-addass 7933 ax-i2m1 7936 ax-0lt1 7937 ax-0id 7939 ax-rnegex 7940 ax-pre-ltirr 7943 ax-pre-lttrn 7945 ax-pre-ltadd 7947 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-ov 5895 df-oprab 5896 df-mpo 5897 df-pnf 8014 df-mnf 8015 df-ltxr 8017 df-inn 8940 df-2 8998 df-3 8999 df-4 9000 df-5 9001 df-6 9002 df-7 9003 df-8 9004 df-ndx 12490 df-slot 12491 df-base 12493 df-sets 12494 df-iress 12495 df-mulr 12576 df-sca 12578 df-vsca 12579 df-ip 12580 df-sra 13719 |
This theorem is referenced by: sratopng 13731 sraring 13733 sralmod 13734 sralmod0g 13735 rlmbasg 13739 |
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