Theorem rngcsectALTV 48774

Description: A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)

Hypotheses

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)

Assertion

Ref	Expression
rngcsectALTV	⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))

Proof of Theorem rngcsectALTV

Step	Hyp	Ref	Expression
1		rngcsectALTV.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2739	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2739	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2739	. . 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 48772	. . . 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 17712	. 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 48767	. . . . . . 7 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 RngHom 𝑌))
14	13	eleq2d 2825	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RngHom 𝑌)))
15	7, 1, 6, 2, 11, 10	rngchomALTV 48767	. . . . . . 7 ⊢ (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 RngHom 𝑋))
16	15	eleq2d 2825	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ↔ 𝐺 ∈ (𝑌 RngHom 𝑋)))
17	14, 16	anbi12d 638	. . . . 5 ⊢ (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))))
18	17	anbi1d 637	. . . 4 ⊢ (𝜑 → (((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
19	6	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) → 𝑈 ∈ 𝑉)
20	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) → 𝑋 ∈ 𝐵)
21	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) → 𝑌 ∈ 𝐵)
22		simprl 776	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) → 𝐹 ∈ (𝑋 RngHom 𝑌))
23		simprr 778	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) → 𝐺 ∈ (𝑌 RngHom 𝑋))
24	7, 1, 19, 3, 20, 21, 20, 22, 23	rngccoALTV 48770	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 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 48773	. . . . . . 7 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸))
27	26	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸))
28	24, 27	eqeq12d 2755	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋))) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ↔ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))
29	28	pm5.32da 584	. . . 4 ⊢ (𝜑 → (((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))
30	18, 29	bitrd 280	. . 3 ⊢ (𝜑 → (((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))
31		df-3an 1094	. . 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 1094	. . 3 ⊢ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)) ↔ ((𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))
33	30, 31, 32	3bitr4g 315	. 2 ⊢ (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))
34	12, 33	bitrd 280	1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ⟨cop 4562   class class class wbr 5073   I cid 5513   ↾ cres 5621   ∘ ccom 5623  ‘cfv 6486  (class class class)co 7357  Basecbs 17171  Hom chom 17223  compcco 17224  Catccat 17622  Idccid 17623  Sectcsect 17703   RngHom crnghm 20406  RngCatALTVcrngcALTV 48762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-er 8634  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-n0 12430  df-z 12517  df-dec 12637  df-uz 12781  df-fz 13454  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17172  df-plusg 17225  df-hom 17236  df-cco 17237  df-0g 17396  df-cat 17626  df-cid 17627  df-sect 17706  df-mgm 18600  df-mgmhm 18652  df-sgrp 18679  df-mnd 18695  df-mhm 18743  df-grp 18904  df-ghm 19180  df-abl 19750  df-mgp 20114  df-rng 20126  df-rnghm 20408  df-rngcALTV 48763

This theorem is referenced by:  rngcinvALTV  48775