Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > vscaid | Structured version Visualization version GIF version |
Description: Utility theorem: index-independent form of scalar product df-vsca 16570. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
vscaid | ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-vsca 16570 | . 2 ⊢ ·𝑠 = Slot 6 | |
2 | 6nn 11714 | . 2 ⊢ 6 ∈ ℕ | |
3 | 1, 2 | ndxid 16497 | 1 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ‘cfv 6348 6c6 11684 ndxcnx 16468 Slot cslot 16470 ·𝑠 cvsca 16557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-1cn 10583 ax-addcl 10585 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-ndx 16474 df-slot 16475 df-vsca 16570 |
This theorem is referenced by: lmodvsca 16628 ipsvsca 16636 phlvsca 16645 prdsvsca 16721 imasvsca 16781 rmodislmod 19631 sravsca 19883 psrvscafval 20098 zlmvsca 20597 matvsca 20953 algvsca 39660 |
Copyright terms: Public domain | W3C validator |