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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 2819 | . . 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 2819 | . . 3 ⊢ (Inv‘(oppCat‘𝐶)) = (Inv‘(oppCat‘𝐶)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | oppcinv 17042 | . 2 ⊢ (𝜑 → (𝑌(Inv‘(oppCat‘𝐶))𝑋) = (𝑋𝑁𝑌)) |
9 | 2, 1 | oppcbas 16980 | . . 3 ⊢ 𝐵 = (Base‘(oppCat‘𝐶)) |
10 | eqid 2819 | . . 3 ⊢ (Mono‘(oppCat‘𝐶)) = (Mono‘(oppCat‘𝐶)) | |
11 | eqid 2819 | . . 3 ⊢ (Sect‘(oppCat‘𝐶)) = (Sect‘(oppCat‘𝐶)) | |
12 | 2 | oppccat 16984 | . . . 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 17000 | . . . 4 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝐶))𝑋) = (𝑋𝐸𝑌)) |
17 | 14, 16 | eleqtrrd 2914 | . . 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 17040 | . . . 4 ⊢ (𝜑 → (𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹 ↔ 𝐹(𝑋𝑆𝑌)𝐺)) |
21 | 18, 20 | mpbird 259 | . . 3 ⊢ (𝜑 → 𝐺(𝑋(Sect‘(oppCat‘𝐶))𝑌)𝐹) |
22 | 9, 10, 11, 13, 4, 5, 7, 17, 21 | monsect 17045 | . 2 ⊢ (𝜑 → 𝐹(𝑌(Inv‘(oppCat‘𝐶))𝑋)𝐺) |
23 | 8, 22 | breqdi 5072 | 1 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 Catccat 16927 oppCatcoppc 16973 Monocmon 16990 Epicepi 16991 Sectcsect 17006 Invcinv 17007 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-tpos 7884 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-hom 16581 df-cco 16582 df-cat 16931 df-cid 16932 df-oppc 16974 df-mon 16992 df-epi 16993 df-sect 17009 df-inv 17010 |
This theorem is referenced by: (None) |
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