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Mirrors > Home > MPE Home > Th. List > slotsinbpsd | Structured version Visualization version GIF version |
Description: The slots Base, +g, ·𝑠 and dist are different from the slot Itv. Formerly part of ttglem 27861 and proofs using it. (Contributed by AV, 29-Oct-2024.) |
Ref | Expression |
---|---|
slotsinbpsd | ⊢ (((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) ∧ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itvndx 27421 | . . . 4 ⊢ (Itv‘ndx) = ;16 | |
2 | 1re 11162 | . . . . . 6 ⊢ 1 ∈ ℝ | |
3 | 1nn 12171 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
4 | 6nn0 12441 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
5 | 1nn0 12436 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 12764 | . . . . . . 7 ⊢ 1 < ;10 | |
7 | 3, 4, 5, 6 | declti 12663 | . . . . . 6 ⊢ 1 < ;16 |
8 | 2, 7 | gtneii 11274 | . . . . 5 ⊢ ;16 ≠ 1 |
9 | basendx 17099 | . . . . 5 ⊢ (Base‘ndx) = 1 | |
10 | 8, 9 | neeqtrri 3018 | . . . 4 ⊢ ;16 ≠ (Base‘ndx) |
11 | 1, 10 | eqnetri 3015 | . . 3 ⊢ (Itv‘ndx) ≠ (Base‘ndx) |
12 | 2re 12234 | . . . . . 6 ⊢ 2 ∈ ℝ | |
13 | 2nn0 12437 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
14 | 2lt10 12763 | . . . . . . 7 ⊢ 2 < ;10 | |
15 | 3, 4, 13, 14 | declti 12663 | . . . . . 6 ⊢ 2 < ;16 |
16 | 12, 15 | gtneii 11274 | . . . . 5 ⊢ ;16 ≠ 2 |
17 | plusgndx 17166 | . . . . 5 ⊢ (+g‘ndx) = 2 | |
18 | 16, 17 | neeqtrri 3018 | . . . 4 ⊢ ;16 ≠ (+g‘ndx) |
19 | 1, 18 | eqnetri 3015 | . . 3 ⊢ (Itv‘ndx) ≠ (+g‘ndx) |
20 | 11, 19 | pm3.2i 472 | . 2 ⊢ ((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) |
21 | 6re 12250 | . . . . . 6 ⊢ 6 ∈ ℝ | |
22 | 6lt10 12759 | . . . . . . 7 ⊢ 6 < ;10 | |
23 | 3, 4, 4, 22 | declti 12663 | . . . . . 6 ⊢ 6 < ;16 |
24 | 21, 23 | gtneii 11274 | . . . . 5 ⊢ ;16 ≠ 6 |
25 | vscandx 17207 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) = 6 | |
26 | 24, 25 | neeqtrri 3018 | . . . 4 ⊢ ;16 ≠ ( ·𝑠 ‘ndx) |
27 | 1, 26 | eqnetri 3015 | . . 3 ⊢ (Itv‘ndx) ≠ ( ·𝑠 ‘ndx) |
28 | 2nn 12233 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
29 | 5, 28 | decnncl 12645 | . . . . . . 7 ⊢ ;12 ∈ ℕ |
30 | 29 | nnrei 12169 | . . . . . 6 ⊢ ;12 ∈ ℝ |
31 | 6nn 12249 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
32 | 2lt6 12344 | . . . . . . 7 ⊢ 2 < 6 | |
33 | 5, 13, 31, 32 | declt 12653 | . . . . . 6 ⊢ ;12 < ;16 |
34 | 30, 33 | gtneii 11274 | . . . . 5 ⊢ ;16 ≠ ;12 |
35 | dsndx 17273 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
36 | 34, 35 | neeqtrri 3018 | . . . 4 ⊢ ;16 ≠ (dist‘ndx) |
37 | 1, 36 | eqnetri 3015 | . . 3 ⊢ (Itv‘ndx) ≠ (dist‘ndx) |
38 | 27, 37 | pm3.2i 472 | . 2 ⊢ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx)) |
39 | 20, 38 | pm3.2i 472 | 1 ⊢ (((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) ∧ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ≠ wne 2944 ‘cfv 6501 1c1 11059 2c2 12215 6c6 12219 ;cdc 12625 ndxcnx 17072 Basecbs 17090 +gcplusg 17140 ·𝑠 cvsca 17144 distcds 17149 Itvcitv 27417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-slot 17061 df-ndx 17073 df-base 17091 df-plusg 17153 df-vsca 17157 df-ds 17162 df-itv 27419 |
This theorem is referenced by: ttgbas 27863 ttgplusg 27865 ttgvsca 27868 ttgds 27870 |
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