Theorem rngccoALTV 49367

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 49366	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
6		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6889	. . . . . 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 8014	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
11	8, 9, 10	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
12	11	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	7, 12	eqtrd 2796	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
14		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
15	13, 14	oveq12d 7438	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) RngHom 𝑧) = (𝑌 RngHom 𝑍))
16	6	fveq2d 6889	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
17		op1stg 8013	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
18	8, 9, 17	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
19	18	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
20	16, 19	eqtrd 2796	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋)
21	20, 13	oveq12d 7438	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((1st ‘𝑣) RngHom (2nd ‘𝑣)) = (𝑋 RngHom 𝑌))
22		eqidd 2762	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
23	15, 21, 22	mpoeq123dv 7495	. . 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 5688	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
25	8, 9, 24	syl2anc 596	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
26		rngccoALTV.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
27		ovex 7453	. . . . 5 ⊢ (𝑌 RngHom 𝑍) ∈ V
28		ovex 7453	. . . . 5 ⊢ (𝑋 RngHom 𝑌) ∈ V
29	27, 28	mpoex 8092	. . . 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 7572	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ (𝑌 RngHom 𝑍), 𝑓 ∈ (𝑋 RngHom 𝑌) ↦ (𝑔 ∘ 𝑓)))
32		simprl 783	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
33		simprr 785	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
34	32, 33	coeq12d 5842	. 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 7941	. . 3 ⊢ ((𝐺 ∈ (𝑌 RngHom 𝑍) ∧ 𝐹 ∈ (𝑋 RngHom 𝑌)) → (𝐺 ∘ 𝐹) ∈ V)
38	35, 36, 37	syl2anc 596	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
39	31, 34, 35, 36, 38	ovmpod 7572	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590   × cxp 5649   ∘ ccom 5655  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000  Basecbs 17387  compcco 17440   RngHom crnghm 20664  RngCatALTVcrngcALTV 49359

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-rngcALTV 49360

This theorem is used by:  rngccatidALTV  49368  rngcsectALTV  49371  rhmsubcALTVlem4  49380