Theorem ringccoALTV 48247

Description: Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ringcbasALTV.c	⊢ 𝐶 = (RingCatALTV‘𝑈)
ringcbasALTV.b	⊢ 𝐵 = (Base‘𝐶)
ringcbasALTV.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
ringccoALTV.o	⊢ · = (comp‘𝐶)
ringccoALTV.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ringccoALTV.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
ringccoALTV.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
ringccoALTV.f	⊢ (𝜑 → 𝐹 ∈ (𝑋 RingHom 𝑌))
ringccoALTV.g	⊢ (𝜑 → 𝐺 ∈ (𝑌 RingHom 𝑍))

Assertion

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

Proof of Theorem ringccoALTV

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

Step	Hyp	Ref	Expression
1		ringcbasALTV.c	. . . 4 ⊢ 𝐶 = (RingCatALTV‘𝑈)
2		ringcbasALTV.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		ringcbasALTV.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		ringccoALTV.o	. . . 4 ⊢ · = (comp‘𝐶)
5	1, 2, 3, 4	ringccofvalALTV 48246	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
6		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6885	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
8		ringccoALTV.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		ringccoALTV.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		op2ndg 8006	. . . . . . . 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 2771	. . . . 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 7428	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑣) RingHom 𝑧) = (𝑌 RingHom 𝑍))
16	6	fveq2d 6885	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
17		op1stg 8005	. . . . . . . 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 2771	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = 𝑋)
21	20, 13	oveq12d 7428	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((1st ‘𝑣) RingHom (2nd ‘𝑣)) = (𝑋 RingHom 𝑌))
22		eqidd 2737	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
23	15, 21, 22	mpoeq123dv 7487	. . 3 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔 ∘ 𝑓)))
24		opelxpi 5696	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
25	8, 9, 24	syl2anc 584	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
26		ringccoALTV.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
27		ovex 7443	. . . . 5 ⊢ (𝑌 RingHom 𝑍) ∈ V
28		ovex 7443	. . . . 5 ⊢ (𝑋 RingHom 𝑌) ∈ V
29	27, 28	mpoex 8083	. . . 4 ⊢ (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔 ∘ 𝑓)) ∈ V
30	29	a1i 11	. . 3 ⊢ (𝜑 → (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔 ∘ 𝑓)) ∈ V)
31	5, 23, 25, 26, 30	ovmpod 7564	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 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 5849	. 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
35		ringccoALTV.g	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌 RingHom 𝑍))
36		ringccoALTV.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋 RingHom 𝑌))
37		coexg 7930	. . 3 ⊢ ((𝐺 ∈ (𝑌 RingHom 𝑍) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌)) → (𝐺 ∘ 𝐹) ∈ V)
38	35, 36, 37	syl2anc 584	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
39	31, 34, 35, 36, 38	ovmpod 7564	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464  ⟨cop 4612   × cxp 5657   ∘ ccom 5663  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7991  2^nd c2nd 7992  Basecbs 17233  compcco 17288   RingHom crh 20434  RingCatALTVcringcALTV 48229

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-slot 17206  df-ndx 17218  df-base 17234  df-hom 17300  df-cco 17301  df-ringcALTV 48230

This theorem is referenced by:  ringccatidALTV  48248  ringcsectALTV  48251  funcringcsetclem9ALTV  48263