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Mirrors > Home > ILE Home > Th. List > basendx | GIF version |
Description: Index value of the base
set extractor.
Use of this theorem is discouraged since the particular value 1 for the index is an implementation detail. It is generally sufficient to work with (Base‘ndx) and use theorems such as baseid 12515 and basendxnn 12517. The main circumstance in which it is necessary to look at indices directly is when showing that a set of indices are disjoint, in proofs such as lmodstrd 12621. Although we have a few theorems such as basendxnplusgndx 12582, we do not intend to add such theorems for every pair of indices (which would be quadradically many in the number of indices). (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Aug-2013.) |
Ref | Expression |
---|---|
basendx | ⊢ (Base‘ndx) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-base 12467 | . 2 ⊢ Base = Slot 1 | |
2 | 1nn 8929 | . 2 ⊢ 1 ∈ ℕ | |
3 | 1, 2 | ndxarg 12484 | 1 ⊢ (Base‘ndx) = 1 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ‘cfv 5216 1c1 7811 ndxcnx 12458 Basecbs 12461 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fv 5224 df-inn 8919 df-ndx 12464 df-slot 12465 df-base 12467 |
This theorem is referenced by: basendxltplusgndx 12571 1strstrg 12574 2strstrg 12576 2strbasg 12577 2stropg 12578 2strstr1g 12579 rngstrg 12592 starvndxnbasendx 12599 scandxnbasendx 12611 vscandxnbasendx 12616 lmodstrd 12621 ipndxnbasendx 12629 basendxlttsetndx 12644 topgrpstrd 12650 basendxltplendx 12658 basendxltdsndx 12669 basendxltunifndx 12679 setsmsbasg 13949 |
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