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Mirrors > Home > ILE Home > Th. List > mulrslid | GIF version |
Description: Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.) |
Ref | Expression |
---|---|
mulrslid | ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mulr 11872 | . 2 ⊢ .r = Slot 3 | |
2 | 3nn 8780 | . 2 ⊢ 3 ∈ ℕ | |
3 | 1, 2 | ndxslid 11821 | 1 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1312 ∈ wcel 1461 ‘cfv 5079 ℕcn 8624 3c3 8676 ndxcnx 11793 Slot cslot 11795 .rcmulr 11859 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-cnex 7630 ax-resscn 7631 ax-1re 7633 ax-addrcl 7636 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-sbc 2877 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-iota 5044 df-fun 5081 df-fv 5087 df-ov 5729 df-inn 8625 df-2 8683 df-3 8684 df-ndx 11799 df-slot 11800 df-mulr 11872 |
This theorem is referenced by: rngmulrg 11914 srngmulrd 11921 ipsmulrd 11940 |
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