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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 2736 | . . 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 2736 | . . 3 ⊢ (Inv‘(oppCat‘𝐶)) = (Inv‘(oppCat‘𝐶)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oppcinv 17239 | . 2 ⊢ (𝜑 → (𝑌(Inv‘(oppCat‘𝐶))𝑋) = (𝑋𝑁𝑌)) |
9 | 2, 1 | oppcbas 17176 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
10 | eqid 2736 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
11 | eqid 2736 | . . 3 ⊢ (Sect‘(oppCat‘𝐶)) = (Sect‘(oppCat‘𝐶)) | |
12 | 2 | oppccat 17180 | . . . 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 17197 | . . . 4 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) |
17 | 14, 16 | eleqtrrd 2834 | . . 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 17237 | . . . 4 ⊢ (𝜑 → (𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹 ↔ 𝐹(𝑋𝑆𝑌)𝐺)) |
21 | 18, 20 | mpbird 260 | . . 3 ⊢ (𝜑 → 𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹) |
22 | 9, 10, 11, 13, 4, 5, 7, 17, 21 | monsect 17242 | . 2 ⊢ (𝜑 → 𝐹(𝑌(Inv‘(oppCat‘𝐶))𝑋)𝐺) |
23 | 8, 22 | breqdi 5054 | 1 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Catccat 17121 oppCatcoppc 17168 Monocmon 17187 Epicepi 17188 Sectcsect 17203 Invcinv 17204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-hom 16773 df-cco 16774 df-cat 17125 df-cid 17126 df-oppc 17169 df-mon 17189 df-epi 17190 df-sect 17206 df-inv 17207 |
This theorem is referenced by: (None) |
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