Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > plusgndxnmulrndx | Unicode version |
Description: The slot for the group (addition) operation is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) |
Ref | Expression |
---|---|
plusgndxnmulrndx |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plusgndx 12052 | . 2 | |
2 | 2re 8790 | . . . 4 | |
3 | 2lt3 8890 | . . . 4 | |
4 | 2, 3 | ltneii 7860 | . . 3 |
5 | mulrndx 12069 | . . 3 | |
6 | 4, 5 | neeqtrri 2337 | . 2 |
7 | 1, 6 | eqnetri 2331 | 1 |
Colors of variables: wff set class |
Syntax hints: wne 2308 cfv 5123 c2 8771 c3 8772 cnx 11956 cplusg 12021 cmulr 12022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-inn 8721 df-2 8779 df-3 8780 df-ndx 11962 df-slot 11963 df-plusg 12034 df-mulr 12035 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |