Theorem ringccofvalALTV 47681

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)

Assertion

Ref	Expression
ringccofvalALTV	⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))

Distinct variable groups:   𝑓,𝑔,𝑣,𝑧   𝑣,𝑈,𝑧   𝜑,𝑣,𝑧   𝑣,𝐵,𝑧

Allowed substitution hints:   𝜑(𝑓,𝑔)   𝐵(𝑓,𝑔)   𝐶(𝑧,𝑣,𝑓,𝑔)   · (𝑧,𝑣,𝑓,𝑔)   𝑈(𝑓,𝑔)   𝑉(𝑧,𝑣,𝑓,𝑔)

Proof of Theorem ringccofvalALTV

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringcbasALTV.c	. . . 4 ⊢ 𝐶 = (RingCatALTV‘𝑈)
2		ringcbasALTV.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		ringcbasALTV.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
4	1, 3, 2	ringcbasALTV 47677	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
5		eqid 2726	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6	1, 3, 2, 5	ringchomfvalALTV 47678	. . . 4 ⊢ (𝜑 → (Hom ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)))
7		eqidd 2727	. . . 4 ⊢ (𝜑 → (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
8	1, 2, 4, 6, 7	ringcvalALTV 47666	. . 3 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
9	8	fveq2d 6905	. 2 ⊢ (𝜑 → (comp‘𝐶) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩}))
10		ringccoALTV.o	. 2 ⊢ · = (comp‘𝐶)
11	3	fvexi 6915	. . . . 5 ⊢ 𝐵 ∈ V
12		sqxpexg 7763	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
13	11, 12	ax-mp 5	. . . 4 ⊢ (𝐵 × 𝐵) ∈ V
14	13, 11	mpoex 8093	. . 3 ⊢ (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) ∈ V
15		catstr 17981	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩} Struct ⟨1, ;15⟩
16		ccoid 17428	. . . 4 ⊢ comp = Slot (comp‘ndx)
17		snsstp3 4827	. . . 4 ⊢ {⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩}
18	15, 16, 17	strfv 17206	. . 3 ⊢ ((𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) ∈ V → (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩}))
19	14, 18	ax-mp 5	. 2 ⊢ (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = (comp‘{⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (Hom ‘𝐶)⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
20	9, 10, 19	3eqtr4g 2791	1 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3462  {ctp 4637  ⟨cop 4639   × cxp 5680   ∘ ccom 5686  ‘cfv 6554  (class class class)co 7424   ∈ cmpo 7426  1^st c1st 8001  2^nd c2nd 8002  1c1 11159  5c5 12322  ;cdc 12729  ndxcnx 17195  Basecbs 17213  Hom chom 17277  compcco 17278   RingHom crh 20451  RingCatALTVcringcALTV 47664

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 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-struct 17149  df-slot 17184  df-ndx 17196  df-base 17214  df-hom 17290  df-cco 17291  df-ringcALTV 47665

This theorem is referenced by:  ringccoALTV  47682