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Mirrors > Home > MPE Home > Th. List > catstr | Structured version Visualization version GIF version |
Description: A category structure is a structure. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
catstr | ⊢ {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉} Struct 〈1, ;15〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 12163 | . 2 ⊢ 1 ∈ ℕ | |
2 | basendx 17091 | . 2 ⊢ (Base‘ndx) = 1 | |
3 | 4nn0 12431 | . . 3 ⊢ 4 ∈ ℕ0 | |
4 | 1nn0 12428 | . . 3 ⊢ 1 ∈ ℕ0 | |
5 | 1lt10 12756 | . . 3 ⊢ 1 < ;10 | |
6 | 1, 3, 4, 5 | declti 12655 | . 2 ⊢ 1 < ;14 |
7 | 4nn 12235 | . . 3 ⊢ 4 ∈ ℕ | |
8 | 4, 7 | decnncl 12637 | . 2 ⊢ ;14 ∈ ℕ |
9 | homndx 17291 | . 2 ⊢ (Hom ‘ndx) = ;14 | |
10 | 5nn 12238 | . . 3 ⊢ 5 ∈ ℕ | |
11 | 4lt5 12329 | . . 3 ⊢ 4 < 5 | |
12 | 4, 3, 10, 11 | declt 12645 | . 2 ⊢ ;14 < ;15 |
13 | 4, 10 | decnncl 12637 | . 2 ⊢ ;15 ∈ ℕ |
14 | ccondx 17293 | . 2 ⊢ (comp‘ndx) = ;15 | |
15 | 1, 2, 6, 8, 9, 12, 13, 14 | strle3 17031 | 1 ⊢ {〈(Base‘ndx), 𝑈〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉} Struct 〈1, ;15〉 |
Colors of variables: wff setvar class |
Syntax hints: {ctp 4590 〈cop 4592 class class class wbr 5105 ‘cfv 6496 1c1 11051 4c4 12209 5c5 12210 ;cdc 12617 Struct cstr 17017 ndxcnx 17064 Basecbs 17082 Hom chom 17143 compcco 17144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-fz 13424 df-struct 17018 df-slot 17053 df-ndx 17065 df-base 17083 df-hom 17156 df-cco 17157 |
This theorem is referenced by: fuccofval 17846 fucbas 17847 fuchom 17848 fuchomOLD 17849 setcbas 17963 setchomfval 17964 setccofval 17967 catcbas 17986 catchomfval 17987 catccofval 17989 estrcbas 18011 estrchomfval 18012 estrccofval 18015 xpchomfval 18066 xpccofval 18069 rngcbasALTV 46252 rngchomfvalALTV 46253 rngccofvalALTV 46256 ringcbasALTV 46315 ringchomfvalALTV 46316 ringccofvalALTV 46319 mndtcbasval 47077 mndtchom 47081 mndtcco 47082 |
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