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Mirrors > Home > ILE Home > Th. List > vscaslid | GIF version |
Description: Slot property of ·𝑠. (Contributed by Jim Kingdon, 5-Feb-2023.) |
Ref | Expression |
---|---|
vscaslid | ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-vsca 12567 | . 2 ⊢ ·𝑠 = Slot 6 | |
2 | 6nn 9097 | . 2 ⊢ 6 ∈ ℕ | |
3 | 1, 2 | ndxslid 12500 | 1 ⊢ ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1363 ∈ wcel 2158 ‘cfv 5228 ℕcn 8932 6c6 8987 ndxcnx 12472 Slot cslot 12474 ·𝑠 cvsca 12554 |
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-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-iota 5190 df-fun 5230 df-fv 5236 df-ov 5891 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-ndx 12478 df-slot 12479 df-vsca 12567 |
This theorem is referenced by: lmodvscad 12640 ipsvscad 12653 ressvscag 12656 prdsex 12735 islmod 13475 scafvalg 13491 scaffng 13493 rmodislmodlem 13534 rmodislmod 13535 lsssn0 13554 lss1d 13567 lssintclm 13568 lspsnel 13601 sraval 13621 sralemg 13622 srascag 13626 sravscag 13627 sraipg 13628 sraex 13630 zlmval 13760 zlmlemg 13761 zlmsca 13765 zlmvscag 13766 |
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