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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 17174 | . . 3 ⊢ (Base‘ndx) = 1 | |
2 | 1re 11230 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 1nn 12239 | . . . . . 6 ⊢ 1 ∈ ℕ | |
4 | 4nn0 12507 | . . . . . 6 ⊢ 4 ∈ ℕ0 | |
5 | 1nn0 12504 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
6 | 1lt10 12832 | . . . . . 6 ⊢ 1 < ;10 | |
7 | 3, 4, 5, 6 | declti 12731 | . . . . 5 ⊢ 1 < ;14 |
8 | 2, 7 | ltneii 11343 | . . . 4 ⊢ 1 ≠ ;14 |
9 | homndx 17377 | . . . 4 ⊢ (Hom ‘ndx) = ;14 | |
10 | 8, 9 | neeqtrri 3009 | . . 3 ⊢ 1 ≠ (Hom ‘ndx) |
11 | 1, 10 | eqnetri 3006 | . 2 ⊢ (Base‘ndx) ≠ (Hom ‘ndx) |
12 | 5nn0 12508 | . . . . . 6 ⊢ 5 ∈ ℕ0 | |
13 | 3, 12, 5, 6 | declti 12731 | . . . . 5 ⊢ 1 < ;15 |
14 | 2, 13 | ltneii 11343 | . . . 4 ⊢ 1 ≠ ;15 |
15 | ccondx 17379 | . . . 4 ⊢ (comp‘ndx) = ;15 | |
16 | 14, 15 | neeqtrri 3009 | . . 3 ⊢ 1 ≠ (comp‘ndx) |
17 | 1, 16 | eqnetri 3006 | . 2 ⊢ (Base‘ndx) ≠ (comp‘ndx) |
18 | 5, 4 | deccl 12708 | . . . . . 6 ⊢ ;14 ∈ ℕ0 |
19 | 18 | nn0rei 12499 | . . . . 5 ⊢ ;14 ∈ ℝ |
20 | 5nn 12314 | . . . . . 6 ⊢ 5 ∈ ℕ | |
21 | 4lt5 12405 | . . . . . 6 ⊢ 4 < 5 | |
22 | 5, 4, 20, 21 | declt 12721 | . . . . 5 ⊢ ;14 < ;15 |
23 | 19, 22 | ltneii 11343 | . . . 4 ⊢ ;14 ≠ ;15 |
24 | 23, 15 | neeqtrri 3009 | . . 3 ⊢ ;14 ≠ (comp‘ndx) |
25 | 9, 24 | eqnetri 3006 | . 2 ⊢ (Hom ‘ndx) ≠ (comp‘ndx) |
26 | 11, 17, 25 | 3pm3.2i 1337 | 1 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1085 ≠ wne 2935 ‘cfv 6542 1c1 11125 4c4 12285 5c5 12286 ;cdc 12693 ndxcnx 17147 Basecbs 17165 Hom chom 17229 compcco 17230 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-slot 17136 df-ndx 17148 df-base 17166 df-hom 17242 df-cco 17243 |
This theorem is referenced by: resshom 17385 ressco 17386 oppchomfval 17679 oppcbas 17684 rescbas 17797 rescco 17801 rescabs 17803 estrreslem1 18112 estrreslem1OLD 18113 estrres 18115 prstcbas 47986 prstchomval 47993 |
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