ringccoALTV - Mathbox for Alexander van der Vekens

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Theorem ringccoALTV 47280

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 47279	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RingHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
6		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
7	6	fveq2d 6895	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2^nd ‘𝑣) = (2^nd ‘⟨𝑋, 𝑌⟩))
8		ringccoALTV.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		ringccoALTV.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		op2ndg 7998	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
11	8, 9, 10	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	7, 12	eqtrd 2767	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2^nd ‘𝑣) = 𝑌)
14		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
15	13, 14	oveq12d 7432	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2^nd ‘𝑣) RingHom 𝑧) = (𝑌 RingHom 𝑍))
16	6	fveq2d 6895	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1^st ‘𝑣) = (1^st ‘⟨𝑋, 𝑌⟩))
17		op1stg 7997	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
18	8, 9, 17	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
20	16, 19	eqtrd 2767	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1^st ‘𝑣) = 𝑋)
21	20, 13	oveq12d 7432	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((1^st ‘𝑣) RingHom (2^nd ‘𝑣)) = (𝑋 RingHom 𝑌))
22		eqidd 2728	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
23	15, 21, 22	mpoeq123dv 7489	. . 3 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2^nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RingHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔 ∘ 𝑓)))
24		opelxpi 5709	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
25	8, 9, 24	syl2anc 583	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
26		ringccoALTV.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
27		ovex 7447	. . . . 5 ⊢ (𝑌 RingHom 𝑍) ∈ V
28		ovex 7447	. . . . 5 ⊢ (𝑋 RingHom 𝑌) ∈ V
29	27, 28	mpoex 8076	. . . 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 7565	. 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 5861	. 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 7929	. . 3 ⊢ ((𝐺 ∈ (𝑌 RingHom 𝑍) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌)) → (𝐺 ∘ 𝐹) ∈ V)
38	35, 36, 37	syl2anc 583	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
39	31, 34, 35, 36, 38	ovmpod 7565	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ⟨cop 4630 × cxp 5670 ∘ ccom 5676 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 1^st c1st 7983 2^nd c2nd 7984 Basecbs 17165 compcco 17230 RingHom crh 20390 RingCatALTVcringcALTV 47262

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-struct 17101 df-slot 17136 df-ndx 17148 df-base 17166 df-hom 17242 df-cco 17243 df-ringcALTV 47263

This theorem is referenced by: ringccatidALTV 47281 ringcsectALTV 47284 funcringcsetclem9ALTV 47296

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