Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > rngcsectALTV | Structured version Visualization version GIF version |
Description: A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rngcsectALTV.c | ⊢ 𝐶 = (RngCatALTV‘𝑈) |
rngcsectALTV.b | ⊢ 𝐵 = (Base‘𝐶) |
rngcsectALTV.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcsectALTV.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
rngcsectALTV.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
rngcsectALTV.e | ⊢ 𝐸 = (Base‘𝑋) |
rngcsectALTV.n | ⊢ 𝑆 = (Sect‘𝐶) |
Ref | Expression |
---|---|
rngcsectALTV | ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcsectALTV.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2821 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
3 | eqid 2821 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
4 | eqid 2821 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
5 | rngcsectALTV.n | . . 3 ⊢ 𝑆 = (Sect‘𝐶) | |
6 | rngcsectALTV.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
7 | rngcsectALTV.c | . . . . 5 ⊢ 𝐶 = (RngCatALTV‘𝑈) | |
8 | 7 | rngccatALTV 44255 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
10 | rngcsectALTV.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | rngcsectALTV.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
12 | 1, 2, 3, 4, 5, 9, 10, 11 | issect 17017 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))) |
13 | 7, 1, 6, 2, 10, 11 | rngchomALTV 44250 | . . . . . . 7 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 RngHomo 𝑌)) |
14 | 13 | eleq2d 2898 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RngHomo 𝑌))) |
15 | 7, 1, 6, 2, 11, 10 | rngchomALTV 44250 | . . . . . . 7 ⊢ (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 RngHomo 𝑋)) |
16 | 15 | eleq2d 2898 | . . . . . 6 ⊢ (𝜑 → (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ↔ 𝐺 ∈ (𝑌 RngHomo 𝑋))) |
17 | 14, 16 | anbi12d 632 | . . . . 5 ⊢ (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)))) |
18 | 17 | anbi1d 631 | . . . 4 ⊢ (𝜑 → (((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))) |
19 | 6 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑈 ∈ 𝑉) |
20 | 10 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑋 ∈ 𝐵) |
21 | 11 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑌 ∈ 𝐵) |
22 | simprl 769 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝐹 ∈ (𝑋 RngHomo 𝑌)) | |
23 | simprr 771 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝐺 ∈ (𝑌 RngHomo 𝑋)) | |
24 | 7, 1, 19, 3, 20, 21, 20, 22, 23 | rngccoALTV 44253 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = (𝐺 ∘ 𝐹)) |
25 | rngcsectALTV.e | . . . . . . . 8 ⊢ 𝐸 = (Base‘𝑋) | |
26 | 7, 1, 4, 6, 10, 25 | rngcidALTV 44256 | . . . . . . 7 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸)) |
27 | 26 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸)) |
28 | 24, 27 | eqeq12d 2837 | . . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → ((𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ↔ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))) |
29 | 28 | pm5.32da 581 | . . . 4 ⊢ (𝜑 → (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))) |
30 | 18, 29 | bitrd 281 | . . 3 ⊢ (𝜑 → (((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))) |
31 | df-3an 1085 | . . 3 ⊢ ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))) | |
32 | df-3an 1085 | . . 3 ⊢ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))) | |
33 | 30, 31, 32 | 3bitr4g 316 | . 2 ⊢ (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))) |
34 | 12, 33 | bitrd 281 | 1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 〈cop 4566 class class class wbr 5058 I cid 5453 ↾ cres 5551 ∘ ccom 5553 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Hom chom 16570 compcco 16571 Catccat 16929 Idccid 16930 Sectcsect 17008 RngHomo crngh 44150 RngCatALTVcrngcALTV 44223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-hom 16583 df-cco 16584 df-0g 16709 df-cat 16933 df-cid 16934 df-sect 17011 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-grp 18100 df-ghm 18350 df-abl 18903 df-mgp 19234 df-mgmhm 44040 df-rng0 44140 df-rnghomo 44152 df-rngcALTV 44225 |
This theorem is referenced by: rngcinvALTV 44258 |
Copyright terms: Public domain | W3C validator |