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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 2772 | . . 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 2772 | . . 3 ⊢ (Inv‘(oppCat‘𝐶)) = (Inv‘(oppCat‘𝐶)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oppcinv 16898 | . 2 ⊢ (𝜑 → (𝑌(Inv‘(oppCat‘𝐶))𝑋) = (𝑋𝑁𝑌)) |
9 | 2, 1 | oppcbas 16836 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
10 | eqid 2772 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
11 | eqid 2772 | . . 3 ⊢ (Sect‘(oppCat‘𝐶)) = (Sect‘(oppCat‘𝐶)) | |
12 | 2 | oppccat 16840 | . . . 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 16856 | . . . 4 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) |
17 | 14, 16 | eleqtrrd 2863 | . . 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 16896 | . . . 4 ⊢ (𝜑 → (𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹 ↔ 𝐹(𝑋𝑆𝑌)𝐺)) |
21 | 18, 20 | mpbird 249 | . . 3 ⊢ (𝜑 → 𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹) |
22 | 9, 10, 11, 13, 4, 5, 7, 17, 21 | monsect 16901 | . 2 ⊢ (𝜑 → 𝐹(𝑌(Inv‘(oppCat‘𝐶))𝑋)𝐺) |
23 | 8, 22 | breqdi 4938 | 1 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2048 class class class wbr 4923 ‘cfv 6182 (class class class)co 6970 Basecbs 16329 Catccat 16783 oppCatcoppc 16829 Monocmon 16846 Epicepi 16847 Sectcsect 16862 Invcinv 16863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-tpos 7688 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-hom 16435 df-cco 16436 df-cat 16787 df-cid 16788 df-oppc 16830 df-mon 16848 df-epi 16849 df-sect 16865 df-inv 16866 |
This theorem is referenced by: (None) |
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