Theorem rngccoALTV 48269

Description: Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)

Hypotheses

Ref	Expression
rngcbasALTV.c	⊢ 𝐶 = (RngCatALTV‘𝑈)
rngcbasALTV.b	⊢ 𝐵 = (Base‘𝐶)
rngcbasALTV.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rngccofvalALTV.o	⊢ · = (comp‘𝐶)
rngccoALTV.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
rngccoALTV.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
rngccoALTV.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
rngccoALTV.f	⊢ (𝜑 → 𝐹 ∈ (𝑋 RngHom 𝑌))
rngccoALTV.g	⊢ (𝜑 → 𝐺 ∈ (𝑌 RngHom 𝑍))

Assertion

Ref	Expression
rngccoALTV	⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Proof of Theorem rngccoALTV

Dummy variables 𝑓 𝑔 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngcbasALTV.c	. . . 4 ⊢ 𝐶 = (RngCatALTV‘𝑈)
2		rngcbasALTV.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		rngcbasALTV.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		rngccofvalALTV.o	. . . 4 ⊢ · = (comp‘𝐶)
5	1, 2, 3, 4	rngccofvalALTV 48268	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
6		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6820	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
8		rngccoALTV.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		rngccoALTV.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		op2ndg 7928	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
11	8, 9, 10	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	7, 12	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
14		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
15	13, 14	oveq12d 7358	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) RngHom 𝑧) = (𝑌 RngHom 𝑍))
16	6	fveq2d 6820	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
17		op1stg 7927	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
18	8, 9, 17	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
20	16, 19	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋)
21	20, 13	oveq12d 7358	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((1st ‘𝑣) RngHom (2nd ‘𝑣)) = (𝑋 RngHom 𝑌))
22		eqidd 2730	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
23	15, 21, 22	mpoeq123dv 7415	. . 3 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑌 RngHom 𝑍), 𝑓 ∈ (𝑋 RngHom 𝑌) ↦ (𝑔 ∘ 𝑓)))
24		opelxpi 5650	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
25	8, 9, 24	syl2anc 584	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
26		rngccoALTV.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
27		ovex 7373	. . . . 5 ⊢ (𝑌 RngHom 𝑍) ∈ V
28		ovex 7373	. . . . 5 ⊢ (𝑋 RngHom 𝑌) ∈ V
29	27, 28	mpoex 8005	. . . 4 ⊢ (𝑔 ∈ (𝑌 RngHom 𝑍), 𝑓 ∈ (𝑋 RngHom 𝑌) ↦ (𝑔 ∘ 𝑓)) ∈ V
30	29	a1i 11	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝑌 RngHom 𝑍), 𝑓 ∈ (𝑋 RngHom 𝑌) ↦ (𝑔 ∘ 𝑓)) ∈ V)
31	5, 23, 25, 26, 30	ovmpod 7492	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ (𝑌 RngHom 𝑍), 𝑓 ∈ (𝑋 RngHom 𝑌) ↦ (𝑔 ∘ 𝑓)))
32		simprl 770	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
33		simprr 772	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
34	32, 33	coeq12d 5801	. 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
35		rngccoALTV.g	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌 RngHom 𝑍))
36		rngccoALTV.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋 RngHom 𝑌))
37		coexg 7853	. . 3 ⊢ ((𝐺 ∈ (𝑌 RngHom 𝑍) ∧ 𝐹 ∈ (𝑋 RngHom 𝑌)) → (𝐺 ∘ 𝐹) ∈ V)
38	35, 36, 37	syl2anc 584	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
39	31, 34, 35, 36, 38	ovmpod 7492	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3433  ⟨cop 4579   × cxp 5611   ∘ ccom 5617  ‘cfv 6476  (class class class)co 7340   ∈ cmpo 7342  1^st c1st 7913  2^nd c2nd 7914  Basecbs 17107  compcco 17160   RngHom crnghm 20306  RngCatALTVcrngcALTV 48261

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5214  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662  ax-cnex 11053  ax-resscn 11054  ax-1cn 11055  ax-icn 11056  ax-addcl 11057  ax-addrcl 11058  ax-mulcl 11059  ax-mulrcl 11060  ax-mulcom 11061  ax-addass 11062  ax-mulass 11063  ax-distr 11064  ax-i2m1 11065  ax-1ne0 11066  ax-1rid 11067  ax-rnegex 11068  ax-rrecex 11069  ax-cnre 11070  ax-pre-lttri 11071  ax-pre-lttrn 11072  ax-pre-ltadd 11073  ax-pre-mulgt0 11074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-tr 5196  df-id 5508  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5566  df-we 5568  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7297  df-ov 7343  df-oprab 7344  df-mpo 7345  df-om 7791  df-1st 7915  df-2nd 7916  df-frecs 8205  df-wrecs 8236  df-recs 8285  df-rdg 8323  df-1o 8379  df-er 8616  df-en 8864  df-dom 8865  df-sdom 8866  df-fin 8867  df-pnf 11139  df-mnf 11140  df-xr 11141  df-ltxr 11142  df-le 11143  df-sub 11337  df-neg 11338  df-nn 12117  df-2 12179  df-3 12180  df-4 12181  df-5 12182  df-6 12183  df-7 12184  df-8 12185  df-9 12186  df-n0 12373  df-z 12460  df-dec 12580  df-uz 12724  df-fz 13399  df-struct 17045  df-slot 17080  df-ndx 17092  df-base 17108  df-hom 17172  df-cco 17173  df-rngcALTV 48262

This theorem is referenced by:  rngccatidALTV  48270  rngcsectALTV  48273  rhmsubcALTVlem4  48282