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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 28636 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 28196 | . . . 4 ⊢ (Itv‘ndx) = ;16 | |
2 | 1re 11218 | . . . . . 6 ⊢ 1 ∈ ℝ | |
3 | 1nn 12227 | . . . . . . 7 ⊢ 1 ∈ ℕ | |
4 | 6nn0 12497 | . . . . . . 7 ⊢ 6 ∈ ℕ0 | |
5 | 1nn0 12492 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 12820 | . . . . . . 7 ⊢ 1 < ;10 | |
7 | 3, 4, 5, 6 | declti 12719 | . . . . . 6 ⊢ 1 < ;16 |
8 | 2, 7 | gtneii 11330 | . . . . 5 ⊢ ;16 ≠ 1 |
9 | basendx 17162 | . . . . 5 ⊢ (Base‘ndx) = 1 | |
10 | 8, 9 | neeqtrri 3008 | . . . 4 ⊢ ;16 ≠ (Base‘ndx) |
11 | 1, 10 | eqnetri 3005 | . . 3 ⊢ (Itv‘ndx) ≠ (Base‘ndx) |
12 | 2re 12290 | . . . . . 6 ⊢ 2 ∈ ℝ | |
13 | 2nn0 12493 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
14 | 2lt10 12819 | . . . . . . 7 ⊢ 2 < ;10 | |
15 | 3, 4, 13, 14 | declti 12719 | . . . . . 6 ⊢ 2 < ;16 |
16 | 12, 15 | gtneii 11330 | . . . . 5 ⊢ ;16 ≠ 2 |
17 | plusgndx 17232 | . . . . 5 ⊢ (+g‘ndx) = 2 | |
18 | 16, 17 | neeqtrri 3008 | . . . 4 ⊢ ;16 ≠ (+g‘ndx) |
19 | 1, 18 | eqnetri 3005 | . . 3 ⊢ (Itv‘ndx) ≠ (+g‘ndx) |
20 | 11, 19 | pm3.2i 470 | . 2 ⊢ ((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) |
21 | 6re 12306 | . . . . . 6 ⊢ 6 ∈ ℝ | |
22 | 6lt10 12815 | . . . . . . 7 ⊢ 6 < ;10 | |
23 | 3, 4, 4, 22 | declti 12719 | . . . . . 6 ⊢ 6 < ;16 |
24 | 21, 23 | gtneii 11330 | . . . . 5 ⊢ ;16 ≠ 6 |
25 | vscandx 17273 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) = 6 | |
26 | 24, 25 | neeqtrri 3008 | . . . 4 ⊢ ;16 ≠ ( ·𝑠 ‘ndx) |
27 | 1, 26 | eqnetri 3005 | . . 3 ⊢ (Itv‘ndx) ≠ ( ·𝑠 ‘ndx) |
28 | 2nn 12289 | . . . . . . . 8 ⊢ 2 ∈ ℕ | |
29 | 5, 28 | decnncl 12701 | . . . . . . 7 ⊢ ;12 ∈ ℕ |
30 | 29 | nnrei 12225 | . . . . . 6 ⊢ ;12 ∈ ℝ |
31 | 6nn 12305 | . . . . . . 7 ⊢ 6 ∈ ℕ | |
32 | 2lt6 12400 | . . . . . . 7 ⊢ 2 < 6 | |
33 | 5, 13, 31, 32 | declt 12709 | . . . . . 6 ⊢ ;12 < ;16 |
34 | 30, 33 | gtneii 11330 | . . . . 5 ⊢ ;16 ≠ ;12 |
35 | dsndx 17339 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
36 | 34, 35 | neeqtrri 3008 | . . . 4 ⊢ ;16 ≠ (dist‘ndx) |
37 | 1, 36 | eqnetri 3005 | . . 3 ⊢ (Itv‘ndx) ≠ (dist‘ndx) |
38 | 27, 37 | pm3.2i 470 | . 2 ⊢ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx)) |
39 | 20, 38 | pm3.2i 470 | 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 395 ≠ wne 2934 ‘cfv 6537 1c1 11113 2c2 12271 6c6 12275 ;cdc 12681 ndxcnx 17135 Basecbs 17153 +gcplusg 17206 ·𝑠 cvsca 17210 distcds 17215 Itvcitv 28192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-vsca 17223 df-ds 17228 df-itv 28194 |
This theorem is referenced by: ttgbas 28638 ttgplusg 28640 ttgvsca 28643 ttgds 28645 |
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