![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > basendxnmulrndx | GIF version |
Description: The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) |
Ref | Expression |
---|---|
basendxnmulrndx | ⊢ (Base‘ndx) ≠ (.r‘ndx) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-base 12462 | . . 3 ⊢ Base = Slot 1 | |
2 | 1nn 8928 | . . 3 ⊢ 1 ∈ ℕ | |
3 | 1, 2 | ndxarg 12479 | . 2 ⊢ (Base‘ndx) = 1 |
4 | 1re 7955 | . . . 4 ⊢ 1 ∈ ℝ | |
5 | 1lt3 9088 | . . . 4 ⊢ 1 < 3 | |
6 | 4, 5 | ltneii 8052 | . . 3 ⊢ 1 ≠ 3 |
7 | mulrndx 12582 | . . 3 ⊢ (.r‘ndx) = 3 | |
8 | 6, 7 | neeqtrri 2376 | . 2 ⊢ 1 ≠ (.r‘ndx) |
9 | 3, 8 | eqnetri 2370 | 1 ⊢ (Base‘ndx) ≠ (.r‘ndx) |
Colors of variables: wff set class |
Syntax hints: ≠ wne 2347 ‘cfv 5216 1c1 7811 3c3 8969 ndxcnx 12453 Basecbs 12456 .rcmulr 12531 |
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-in1 614 ax-in2 615 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-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 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-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 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-ov 5877 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-mulr 12544 |
This theorem is referenced by: ressmulrg 12597 imasbas 12710 imasmulr 12712 opprbasg 13200 |
Copyright terms: Public domain | W3C validator |