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| Mirrors > Home > MPE Home > Th. List > slotsdifunifndx | Structured version Visualization version GIF version | ||
| Description: The index of the slot for the uniform set is not the index of other slots. Formerly part of proof for cnfldfunALT 21379. (Contributed by AV, 10-Nov-2024.) |
| Ref | Expression |
|---|---|
| slotsdifunifndx | ⊢ (((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2re 12340 | . . . . 5 ⊢ 2 ∈ ℝ | |
| 2 | 1nn 12277 | . . . . . 6 ⊢ 1 ∈ ℕ | |
| 3 | 3nn0 12544 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
| 4 | 2nn0 12543 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
| 5 | 2lt10 12871 | . . . . . 6 ⊢ 2 < ;10 | |
| 6 | 2, 3, 4, 5 | declti 12771 | . . . . 5 ⊢ 2 < ;13 |
| 7 | 1, 6 | ltneii 11374 | . . . 4 ⊢ 2 ≠ ;13 |
| 8 | plusgndx 17323 | . . . . 5 ⊢ (+g‘ndx) = 2 | |
| 9 | unifndx 17439 | . . . . 5 ⊢ (UnifSet‘ndx) = ;13 | |
| 10 | 8, 9 | neeq12i 3007 | . . . 4 ⊢ ((+g‘ndx) ≠ (UnifSet‘ndx) ↔ 2 ≠ ;13) |
| 11 | 7, 10 | mpbir 231 | . . 3 ⊢ (+g‘ndx) ≠ (UnifSet‘ndx) |
| 12 | 3re 12346 | . . . . 5 ⊢ 3 ∈ ℝ | |
| 13 | 3lt10 12870 | . . . . . 6 ⊢ 3 < ;10 | |
| 14 | 2, 3, 3, 13 | declti 12771 | . . . . 5 ⊢ 3 < ;13 |
| 15 | 12, 14 | ltneii 11374 | . . . 4 ⊢ 3 ≠ ;13 |
| 16 | mulrndx 17337 | . . . . 5 ⊢ (.r‘ndx) = 3 | |
| 17 | 16, 9 | neeq12i 3007 | . . . 4 ⊢ ((.r‘ndx) ≠ (UnifSet‘ndx) ↔ 3 ≠ ;13) |
| 18 | 15, 17 | mpbir 231 | . . 3 ⊢ (.r‘ndx) ≠ (UnifSet‘ndx) |
| 19 | 4re 12350 | . . . . 5 ⊢ 4 ∈ ℝ | |
| 20 | 4nn0 12545 | . . . . . 6 ⊢ 4 ∈ ℕ0 | |
| 21 | 4lt10 12869 | . . . . . 6 ⊢ 4 < ;10 | |
| 22 | 2, 3, 20, 21 | declti 12771 | . . . . 5 ⊢ 4 < ;13 |
| 23 | 19, 22 | ltneii 11374 | . . . 4 ⊢ 4 ≠ ;13 |
| 24 | starvndx 17346 | . . . . 5 ⊢ (*𝑟‘ndx) = 4 | |
| 25 | 24, 9 | neeq12i 3007 | . . . 4 ⊢ ((*𝑟‘ndx) ≠ (UnifSet‘ndx) ↔ 4 ≠ ;13) |
| 26 | 23, 25 | mpbir 231 | . . 3 ⊢ (*𝑟‘ndx) ≠ (UnifSet‘ndx) |
| 27 | 11, 18, 26 | 3pm3.2i 1340 | . 2 ⊢ ((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) |
| 28 | 10re 12752 | . . . . 5 ⊢ ;10 ∈ ℝ | |
| 29 | 1nn0 12542 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
| 30 | 0nn0 12541 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
| 31 | 3nn 12345 | . . . . . 6 ⊢ 3 ∈ ℕ | |
| 32 | 3pos 12371 | . . . . . 6 ⊢ 0 < 3 | |
| 33 | 29, 30, 31, 32 | declt 12761 | . . . . 5 ⊢ ;10 < ;13 |
| 34 | 28, 33 | ltneii 11374 | . . . 4 ⊢ ;10 ≠ ;13 |
| 35 | plendx 17410 | . . . . 5 ⊢ (le‘ndx) = ;10 | |
| 36 | 35, 9 | neeq12i 3007 | . . . 4 ⊢ ((le‘ndx) ≠ (UnifSet‘ndx) ↔ ;10 ≠ ;13) |
| 37 | 34, 36 | mpbir 231 | . . 3 ⊢ (le‘ndx) ≠ (UnifSet‘ndx) |
| 38 | 2nn 12339 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
| 39 | 29, 38 | decnncl 12753 | . . . . . 6 ⊢ ;12 ∈ ℕ |
| 40 | 39 | nnrei 12275 | . . . . 5 ⊢ ;12 ∈ ℝ |
| 41 | 2lt3 12438 | . . . . . 6 ⊢ 2 < 3 | |
| 42 | 29, 4, 31, 41 | declt 12761 | . . . . 5 ⊢ ;12 < ;13 |
| 43 | 40, 42 | ltneii 11374 | . . . 4 ⊢ ;12 ≠ ;13 |
| 44 | dsndx 17429 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
| 45 | 44, 9 | neeq12i 3007 | . . . 4 ⊢ ((dist‘ndx) ≠ (UnifSet‘ndx) ↔ ;12 ≠ ;13) |
| 46 | 43, 45 | mpbir 231 | . . 3 ⊢ (dist‘ndx) ≠ (UnifSet‘ndx) |
| 47 | 37, 46 | pm3.2i 470 | . 2 ⊢ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)) |
| 48 | 27, 47 | pm3.2i 470 | 1 ⊢ (((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx))) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∧ w3a 1087 ≠ wne 2940 ‘cfv 6561 0cc0 11155 1c1 11156 2c2 12321 3c3 12322 4c4 12323 ;cdc 12733 ndxcnx 17230 +gcplusg 17297 .rcmulr 17298 *𝑟cstv 17299 lecple 17304 distcds 17306 UnifSetcunif 17307 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-slot 17219 df-ndx 17231 df-plusg 17310 df-mulr 17311 df-starv 17312 df-ple 17317 df-ds 17319 df-unif 17320 |
| This theorem is referenced by: cnfldfunALT 21379 cnfldfunALTOLD 21392 |
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