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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 2824 | . . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
3 | eqid 2824 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
4 | eqid 2824 | . . 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 44268 | . . . 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 17026 | . 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 44263 | . . . . . . 7 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 RngHomo 𝑌)) |
14 | 13 | eleq2d 2901 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RngHomo 𝑌))) |
15 | 7, 1, 6, 2, 11, 10 | rngchomALTV 44263 | . . . . . . 7 ⊢ (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 RngHomo 𝑋)) |
16 | 15 | eleq2d 2901 | . . . . . 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 44266 | . . . . . 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 44269 | . . . . . . 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 2840 | . . . . 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 1536 ∈ wcel 2113 〈cop 4576 class class class wbr 5069 I cid 5462 ↾ cres 5560 ∘ ccom 5562 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 Hom chom 16579 compcco 16580 Catccat 16938 Idccid 16939 Sectcsect 17017 RngHomo crngh 44163 RngCatALTVcrngcALTV 44236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-hom 16592 df-cco 16593 df-0g 16718 df-cat 16942 df-cid 16943 df-sect 17020 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-grp 18109 df-ghm 18359 df-abl 18912 df-mgp 19243 df-mgmhm 44053 df-rng0 44153 df-rnghomo 44165 df-rngcALTV 44238 |
This theorem is referenced by: rngcinvALTV 44271 |
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