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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 2726 | . . 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 2726 | . . 3 ⊢ (Inv‘(oppCat‘𝐶)) = (Inv‘(oppCat‘𝐶)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oppcinv 17791 | . 2 ⊢ (𝜑 → (𝑌(Inv‘(oppCat‘𝐶))𝑋) = (𝑋𝑁𝑌)) |
9 | 2, 1 | oppcbas 17727 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
10 | eqid 2726 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
11 | eqid 2726 | . . 3 ⊢ (Sect‘(oppCat‘𝐶)) = (Sect‘(oppCat‘𝐶)) | |
12 | 2 | oppccat 17732 | . . . 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 17749 | . . . 4 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) |
17 | 14, 16 | eleqtrrd 2829 | . . 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 17789 | . . . 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 17794 | . 2 ⊢ (𝜑 → 𝐹(𝑌(Inv‘(oppCat‘𝐶))𝑋)𝐺) |
23 | 8, 22 | breqdi 5160 | 1 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 class class class wbr 5145 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 Catccat 17672 oppCatcoppc 17719 Monocmon 17739 Epicepi 17740 Sectcsect 17755 Invcinv 17756 |
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-rep 5282 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-rmo 3364 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-1st 7995 df-2nd 7996 df-tpos 8233 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-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-hom 17285 df-cco 17286 df-cat 17676 df-cid 17677 df-oppc 17720 df-mon 17741 df-epi 17742 df-sect 17758 df-inv 17759 |
This theorem is referenced by: (None) |
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