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Mirrors > Home > MPE Home > Th. List > episect | Structured version Visualization version GIF version |
Description: If 𝐹 is an epimorphism and 𝐹 is a section of 𝐺, then 𝐺 is an inverse of 𝐹 and they are both isomorphisms. This is also stated as "an epimorphism which is also a split monomorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
sectepi.b | ⊢ 𝐵 = (Base‘𝐶) |
sectepi.e | ⊢ 𝐸 = (Epi‘𝐶) |
sectepi.s | ⊢ 𝑆 = (Sect‘𝐶) |
sectepi.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
sectepi.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
sectepi.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
episect.n | ⊢ 𝑁 = (Inv‘𝐶) |
episect.1 | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐸𝑌)) |
episect.2 | ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) |
Ref | Expression |
---|---|
episect | ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sectepi.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2737 | . . 3 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶) | |
3 | sectepi.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | sectepi.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
5 | sectepi.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | episect.n | . . 3 ⊢ 𝑁 = (Inv‘𝐶) | |
7 | eqid 2737 | . . 3 ⊢ (Inv‘(oppCat‘𝐶)) = (Inv‘(oppCat‘𝐶)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oppcinv 17623 | . 2 ⊢ (𝜑 → (𝑌(Inv‘(oppCat‘𝐶))𝑋) = (𝑋𝑁𝑌)) |
9 | 2, 1 | oppcbas 17559 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
10 | eqid 2737 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
11 | eqid 2737 | . . 3 ⊢ (Sect‘(oppCat‘𝐶)) = (Sect‘(oppCat‘𝐶)) | |
12 | 2 | oppccat 17564 | . . . 4 ⊢ (𝐶 ∈ Cat → (oppCat‘𝐶) ∈ Cat) |
13 | 3, 12 | syl 17 | . . 3 ⊢ (𝜑 → (oppCat‘𝐶) ∈ Cat) |
14 | episect.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐸𝑌)) | |
15 | sectepi.e | . . . . 5 ⊢ 𝐸 = (Epi‘𝐶) | |
16 | 2, 3, 10, 15 | oppcmon 17581 | . . . 4 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) |
17 | 14, 16 | eleqtrrd 2841 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Mono‘(oppCat‘𝐶))𝑋)) |
18 | episect.2 | . . . 4 ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺) | |
19 | sectepi.s | . . . . 5 ⊢ 𝑆 = (Sect‘𝐶) | |
20 | 1, 2, 3, 5, 4, 19, 11 | oppcsect 17621 | . . . 4 ⊢ (𝜑 → (𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹 ↔ 𝐹(𝑋𝑆𝑌)𝐺)) |
21 | 18, 20 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹) |
22 | 9, 10, 11, 13, 4, 5, 7, 17, 21 | monsect 17626 | . 2 ⊢ (𝜑 → 𝐹(𝑌(Inv‘(oppCat‘𝐶))𝑋)𝐺) |
23 | 8, 22 | breqdi 5118 | 1 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6493 (class class class)co 7351 Basecbs 17043 Catccat 17504 oppCatcoppc 17551 Monocmon 17571 Epicepi 17572 Sectcsect 17587 Invcinv 17588 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-tpos 8149 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-hom 17117 df-cco 17118 df-cat 17508 df-cid 17509 df-oppc 17552 df-mon 17573 df-epi 17574 df-sect 17590 df-inv 17591 |
This theorem is referenced by: (None) |
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