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Mirrors > Home > MPE Home > Th. List > vscaid | Structured version Visualization version GIF version |
Description: Utility theorem: index-independent form of scalar product df-vsca 17278. (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 17278 | . 2 ⊢ ·𝑠 = Slot 6 | |
2 | 6nn 12347 | . 2 ⊢ 6 ∈ ℕ | |
3 | 1, 2 | ndxid 17194 | 1 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ‘cfv 6546 6c6 12317 Slot cslot 17178 ndxcnx 17190 ·𝑠 cvsca 17265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-1cn 11207 ax-addcl 11209 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-slot 17179 df-ndx 17191 df-vsca 17278 |
This theorem is referenced by: lmodvsca 17338 ipsvsca 17350 ressvsca 17353 phlvsca 17359 prdsvsca 17470 imasvsca 17530 rmodislmod 20902 rmodislmodOLD 20903 sravsca 21160 sravscaOLD 21161 zlmvsca 21511 psrvscafval 21953 opsrvsca 22060 matvsca 22405 matvscaOLD 22406 tngvsca 24648 ttgvsca 28808 resvvsca 33216 algvsca 42880 mendvscafval 42888 mnringvscad 43935 |
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