Theorem rngccoALTV 48959

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 48958	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
6		simprl 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6886	. . . . . 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 7999	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
11	8, 9, 10	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
12	11	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	7, 12	eqtrd 2804	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
14		simprr 784	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
15	13, 14	oveq12d 7429	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) RngHom 𝑧) = (𝑌 RngHom 𝑍))
16	6	fveq2d 6886	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
17		op1stg 7998	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
18	8, 9, 17	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
19	18	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
20	16, 19	eqtrd 2804	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋)
21	20, 13	oveq12d 7429	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((1st ‘𝑣) RngHom (2nd ‘𝑣)) = (𝑋 RngHom 𝑌))
22		eqidd 2770	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
23	15, 21, 22	mpoeq123dv 7486	. . 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 5699	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
25	8, 9, 24	syl2anc 595	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
26		rngccoALTV.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
27		ovex 7444	. . . . 5 ⊢ (𝑌 RngHom 𝑍) ∈ V
28		ovex 7444	. . . . 5 ⊢ (𝑋 RngHom 𝑌) ∈ V
29	27, 28	mpoex 8076	. . . 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 7563	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ (𝑌 RngHom 𝑍), 𝑓 ∈ (𝑋 RngHom 𝑌) ↦ (𝑔 ∘ 𝑓)))
32		simprl 782	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
33		simprr 784	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
34	32, 33	coeq12d 5851	. 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 7926	. . 3 ⊢ ((𝐺 ∈ (𝑌 RngHom 𝑍) ∧ 𝐹 ∈ (𝑋 RngHom 𝑌)) → (𝐺 ∘ 𝐹) ∈ V)
38	35, 36, 37	syl2anc 595	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
39	31, 34, 35, 36, 38	ovmpod 7563	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   × cxp 5660   ∘ ccom 5666  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7984  2^nd c2nd 7985  Basecbs 17269  compcco 17322   RngHom crnghm 20516  RngCatALTVcrngcALTV 48951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-struct 17207  df-slot 17242  df-ndx 17254  df-base 17270  df-hom 17334  df-cco 17335  df-rngcALTV 48952

This theorem is referenced by:  rngccatidALTV  48960  rngcsectALTV  48963  rhmsubcALTVlem4  48972