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Mirrors > Home > ILE Home > Th. List > basendxnplusgndx | GIF version |
Description: The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.) |
Ref | Expression |
---|---|
basendxnplusgndx | ⊢ (Base‘ndx) ≠ (+g‘ndx) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-base 12521 | . . 3 ⊢ Base = Slot 1 | |
2 | 1nn 8961 | . . 3 ⊢ 1 ∈ ℕ | |
3 | 1, 2 | ndxarg 12538 | . 2 ⊢ (Base‘ndx) = 1 |
4 | 1ne2 9156 | . . 3 ⊢ 1 ≠ 2 | |
5 | plusgndx 12624 | . . 3 ⊢ (+g‘ndx) = 2 | |
6 | 4, 5 | neeqtrri 2389 | . 2 ⊢ 1 ≠ (+g‘ndx) |
7 | 3, 6 | eqnetri 2383 | 1 ⊢ (Base‘ndx) ≠ (+g‘ndx) |
Colors of variables: wff set class |
Syntax hints: ≠ wne 2360 ‘cfv 5235 1c1 7843 2c2 9001 ndxcnx 12512 Basecbs 12515 +gcplusg 12592 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-i2m1 7947 ax-0lt1 7948 ax-0id 7950 ax-rnegex 7951 ax-pre-ltirr 7954 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-iota 5196 df-fun 5237 df-fv 5243 df-ov 5900 df-pnf 8025 df-mnf 8026 df-ltxr 8028 df-inn 8951 df-2 9009 df-ndx 12518 df-slot 12519 df-base 12521 df-plusg 12605 |
This theorem is referenced by: ressplusgd 12643 imasbas 12787 imasplusg 12788 mgpbasg 13297 mgpress 13302 |
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