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Mirrors > Home > MPE Home > Th. List > slotstnscsi | Structured version Visualization version GIF version |
Description: The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. Formerly part of sralem 20777 and proofs using it. (Contributed by AV, 29-Oct-2024.) |
Ref | Expression |
---|---|
slotstnscsi | ⊢ ((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5re 12294 | . . . 4 ⊢ 5 ∈ ℝ | |
2 | 5lt9 12409 | . . . 4 ⊢ 5 < 9 | |
3 | 1, 2 | gtneii 11321 | . . 3 ⊢ 9 ≠ 5 |
4 | tsetndx 17292 | . . . 4 ⊢ (TopSet‘ndx) = 9 | |
5 | scandx 17254 | . . . 4 ⊢ (Scalar‘ndx) = 5 | |
6 | 4, 5 | neeq12i 3008 | . . 3 ⊢ ((TopSet‘ndx) ≠ (Scalar‘ndx) ↔ 9 ≠ 5) |
7 | 3, 6 | mpbir 230 | . 2 ⊢ (TopSet‘ndx) ≠ (Scalar‘ndx) |
8 | 6re 12297 | . . . 4 ⊢ 6 ∈ ℝ | |
9 | 6lt9 12408 | . . . 4 ⊢ 6 < 9 | |
10 | 8, 9 | gtneii 11321 | . . 3 ⊢ 9 ≠ 6 |
11 | vscandx 17259 | . . . 4 ⊢ ( ·𝑠 ‘ndx) = 6 | |
12 | 4, 11 | neeq12i 3008 | . . 3 ⊢ ((TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ↔ 9 ≠ 6) |
13 | 10, 12 | mpbir 230 | . 2 ⊢ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) |
14 | 8re 12303 | . . . 4 ⊢ 8 ∈ ℝ | |
15 | 8lt9 12406 | . . . 4 ⊢ 8 < 9 | |
16 | 14, 15 | gtneii 11321 | . . 3 ⊢ 9 ≠ 8 |
17 | ipndx 17270 | . . . 4 ⊢ (·𝑖‘ndx) = 8 | |
18 | 4, 17 | neeq12i 3008 | . . 3 ⊢ ((TopSet‘ndx) ≠ (·𝑖‘ndx) ↔ 9 ≠ 8) |
19 | 16, 18 | mpbir 230 | . 2 ⊢ (TopSet‘ndx) ≠ (·𝑖‘ndx) |
20 | 7, 13, 19 | 3pm3.2i 1340 | 1 ⊢ ((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1088 ≠ wne 2941 ‘cfv 6539 5c5 12265 6c6 12266 8c8 12268 9c9 12269 ndxcnx 17121 Scalarcsca 17195 ·𝑠 cvsca 17196 ·𝑖cip 17197 TopSetcts 17198 |
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 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-slot 17110 df-ndx 17122 df-sca 17208 df-vsca 17209 df-ip 17210 df-tset 17211 |
This theorem is referenced by: sratset 20790 tngsca 24139 tngvsca 24141 tngip 24143 zlmtset 32881 |
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