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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 11805 | . . 3 ⊢ Base = Slot 1 | |
2 | 1nn 8638 | . . 3 ⊢ 1 ∈ ℕ | |
3 | 1, 2 | ndxarg 11822 | . 2 ⊢ (Base‘ndx) = 1 |
4 | 1ne2 8827 | . . 3 ⊢ 1 ≠ 2 | |
5 | plusgndx 11892 | . . 3 ⊢ (+g‘ndx) = 2 | |
6 | 4, 5 | neeqtrri 2311 | . 2 ⊢ 1 ≠ (+g‘ndx) |
7 | 3, 6 | eqnetri 2305 | 1 ⊢ (Base‘ndx) ≠ (+g‘ndx) |
Colors of variables: wff set class |
Syntax hints: ≠ wne 2282 ‘cfv 5081 1c1 7545 2c2 8678 ndxcnx 11796 Basecbs 11799 +gcplusg 11861 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7633 ax-resscn 7634 ax-1cn 7635 ax-1re 7636 ax-icn 7637 ax-addcl 7638 ax-addrcl 7639 ax-mulcl 7640 ax-addcom 7642 ax-addass 7644 ax-i2m1 7647 ax-0lt1 7648 ax-0id 7650 ax-rnegex 7651 ax-pre-ltirr 7654 ax-pre-ltadd 7658 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-iota 5046 df-fun 5083 df-fv 5089 df-ov 5731 df-pnf 7723 df-mnf 7724 df-ltxr 7726 df-inn 8628 df-2 8686 df-ndx 11802 df-slot 11803 df-base 11805 df-plusg 11874 |
This theorem is referenced by: (None) |
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