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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-isrvecd | Structured version Visualization version GIF version |
Description: The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.) |
Ref | Expression |
---|---|
bj-isrvecd.scal | β’ (π β (Scalarβπ) = πΎ) |
Ref | Expression |
---|---|
bj-isrvecd | β’ (π β (π β β-Vec β (π β LMod β§ πΎ = βfld))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-isrvec 36043 | . 2 β’ (π β β-Vec β (π β LMod β§ (Scalarβπ) = βfld)) | |
2 | bj-isrvecd.scal | . . . 4 β’ (π β (Scalarβπ) = πΎ) | |
3 | 2 | eqeq1d 2734 | . . 3 β’ (π β ((Scalarβπ) = βfld β πΎ = βfld)) |
4 | 3 | anbi2d 629 | . 2 β’ (π β ((π β LMod β§ (Scalarβπ) = βfld) β (π β LMod β§ πΎ = βfld))) |
5 | 1, 4 | bitrid 282 | 1 β’ (π β (π β β-Vec β (π β LMod β§ πΎ = βfld))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 βcfv 6533 Scalarcsca 17184 LModclmod 20422 βfldcrefld 21092 β-Veccrrvec 36041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-1cn 11152 ax-addcl 11154 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 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-ov 7397 df-om 7840 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-nn 12197 df-2 12259 df-3 12260 df-4 12261 df-5 12262 df-slot 17099 df-ndx 17111 df-sca 17197 df-bj-rvec 36042 |
This theorem is referenced by: bj-isrvec2 36049 |
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