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Mirrors > Home > MPE Home > Th. List > slotsbhcdif | Structured version Visualization version GIF version |
Description: The slots Base, Hom and comp are different. (Contributed by AV, 5-Mar-2020.) (Proof shortened by AV, 28-Oct-2024.) |
Ref | Expression |
---|---|
slotsbhcdif | ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | basendx 17217 | . . 3 ⊢ (Base‘ndx) = 1 | |
2 | 1re 11255 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 1nn 12269 | . . . . . 6 ⊢ 1 ∈ ℕ | |
4 | 4nn0 12537 | . . . . . 6 ⊢ 4 ∈ ℕ0 | |
5 | 1nn0 12534 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 12862 | . . . . . 6 ⊢ 1 < ;10 | |
7 | 3, 4, 5, 6 | declti 12761 | . . . . 5 ⊢ 1 < ;14 |
8 | 2, 7 | ltneii 11368 | . . . 4 ⊢ 1 ≠ ;14 |
9 | homndx 17420 | . . . 4 ⊢ (Hom ‘ndx) = ;14 | |
10 | 8, 9 | neeqtrri 3004 | . . 3 ⊢ 1 ≠ (Hom ‘ndx) |
11 | 1, 10 | eqnetri 3001 | . 2 ⊢ (Base‘ndx) ≠ (Hom ‘ndx) |
12 | 5nn0 12538 | . . . . . 6 ⊢ 5 ∈ ℕ0 | |
13 | 3, 12, 5, 6 | declti 12761 | . . . . 5 ⊢ 1 < ;15 |
14 | 2, 13 | ltneii 11368 | . . . 4 ⊢ 1 ≠ ;15 |
15 | ccondx 17422 | . . . 4 ⊢ (comp‘ndx) = ;15 | |
16 | 14, 15 | neeqtrri 3004 | . . 3 ⊢ 1 ≠ (comp‘ndx) |
17 | 1, 16 | eqnetri 3001 | . 2 ⊢ (Base‘ndx) ≠ (comp‘ndx) |
18 | 5, 4 | deccl 12738 | . . . . . 6 ⊢ ;14 ∈ ℕ0 |
19 | 18 | nn0rei 12529 | . . . . 5 ⊢ ;14 ∈ ℝ |
20 | 5nn 12344 | . . . . . 6 ⊢ 5 ∈ ℕ | |
21 | 4lt5 12435 | . . . . . 6 ⊢ 4 < 5 | |
22 | 5, 4, 20, 21 | declt 12751 | . . . . 5 ⊢ ;14 < ;15 |
23 | 19, 22 | ltneii 11368 | . . . 4 ⊢ ;14 ≠ ;15 |
24 | 23, 15 | neeqtrri 3004 | . . 3 ⊢ ;14 ≠ (comp‘ndx) |
25 | 9, 24 | eqnetri 3001 | . 2 ⊢ (Hom ‘ndx) ≠ (comp‘ndx) |
26 | 11, 17, 25 | 3pm3.2i 1336 | 1 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1084 ≠ wne 2930 ‘cfv 6546 1c1 11150 4c4 12315 5c5 12316 ;cdc 12723 ndxcnx 17190 Basecbs 17208 Hom chom 17272 compcco 17273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 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 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-slot 17179 df-ndx 17191 df-base 17209 df-hom 17285 df-cco 17286 |
This theorem is referenced by: resshom 17428 ressco 17429 oppchomfval 17722 oppcbas 17727 rescbas 17840 rescco 17844 rescabs 17846 estrreslem1 18155 estrreslem1OLD 18156 estrres 18158 prstcbas 48424 prstchomval 48431 |
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