Theorem estrcco 17382

Description: Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)

Hypotheses

Ref	Expression
estrcbas.c	⊢ 𝐶 = (ExtStrCat‘𝑈)
estrcbas.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
estrcco.o	⊢ · = (comp‘𝐶)
estrcco.x	⊢ (𝜑 → 𝑋 ∈ 𝑈)
estrcco.y	⊢ (𝜑 → 𝑌 ∈ 𝑈)
estrcco.z	⊢ (𝜑 → 𝑍 ∈ 𝑈)
estrcco.a	⊢ 𝐴 = (Base‘𝑋)
estrcco.b	⊢ 𝐵 = (Base‘𝑌)
estrcco.d	⊢ 𝐷 = (Base‘𝑍)
estrcco.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
estrcco.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐷)

Assertion

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

Proof of Theorem estrcco

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

Step	Hyp	Ref	Expression
1		estrcbas.c	. . . 4 ⊢ 𝐶 = (ExtStrCat‘𝑈)
2		estrcbas.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		estrcco.o	. . . 4 ⊢ · = (comp‘𝐶)
4	1, 2, 3	estrccofval 17381	. . 3 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
5		fveq2 6672	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (Base‘𝑧) = (Base‘𝑍))
6	5	adantl 484	. . . . . 6 ⊢ ((𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍) → (Base‘𝑧) = (Base‘𝑍))
7	6	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘𝑧) = (Base‘𝑍))
8		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
9	8	fveq2d 6676	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
10		estrcco.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
11		estrcco.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
12		op2ndg 7704	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	10, 11, 12	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
14	13	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
15	9, 14	eqtrd 2858	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
16	15	fveq2d 6676	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(2nd ‘𝑣)) = (Base‘𝑌))
17	7, 16	oveq12d 7176	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))) = ((Base‘𝑍) ↑m (Base‘𝑌)))
18	8	fveq2d 6676	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
19	18	fveq2d 6676	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘𝑣)) = (Base‘(1st ‘⟨𝑋, 𝑌⟩)))
20		op1stg 7703	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
21	10, 11, 20	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
22	21	fveq2d 6676	. . . . . . 7 ⊢ (𝜑 → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
23	22	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
24	19, 23	eqtrd 2858	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘𝑣)) = (Base‘𝑋))
25	16, 24	oveq12d 7176	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) = ((Base‘𝑌) ↑m (Base‘𝑋)))
26		eqidd 2824	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
27	17, 25, 26	mpoeq123dv 7231	. . 3 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↦ (𝑔 ∘ 𝑓)))
28	10, 11	opelxpd 5595	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
29		estrcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
30		ovex 7191	. . . . 5 ⊢ ((Base‘𝑍) ↑m (Base‘𝑌)) ∈ V
31		ovex 7191	. . . . 5 ⊢ ((Base‘𝑌) ↑m (Base‘𝑋)) ∈ V
32	30, 31	mpoex 7779	. . . 4 ⊢ (𝑔 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↦ (𝑔 ∘ 𝑓)) ∈ V
33	32	a1i 11	. . 3 ⊢ (𝜑 → (𝑔 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↦ (𝑔 ∘ 𝑓)) ∈ V)
34	4, 27, 28, 29, 33	ovmpod 7304	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↦ (𝑔 ∘ 𝑓)))
35		simpl 485	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → 𝑔 = 𝐺)
36		simpr 487	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
37	35, 36	coeq12d 5737	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
38	37	adantl 484	. 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
39		estrcco.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
40		estrcco.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑌)
41	40	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝑌))
42	41	eqcomd 2829	. . . . 5 ⊢ (𝜑 → (Base‘𝑌) = 𝐵)
43		estrcco.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝑍)
44	43	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐷 = (Base‘𝑍))
45	44	eqcomd 2829	. . . . 5 ⊢ (𝜑 → (Base‘𝑍) = 𝐷)
46	42, 45	feq23d 6511	. . . 4 ⊢ (𝜑 → (𝐺:(Base‘𝑌)⟶(Base‘𝑍) ↔ 𝐺:𝐵⟶𝐷))
47	39, 46	mpbird 259	. . 3 ⊢ (𝜑 → 𝐺:(Base‘𝑌)⟶(Base‘𝑍))
48		fvexd 6687	. . . 4 ⊢ (𝜑 → (Base‘𝑍) ∈ V)
49		fvexd 6687	. . . 4 ⊢ (𝜑 → (Base‘𝑌) ∈ V)
50	48, 49	elmapd 8422	. . 3 ⊢ (𝜑 → (𝐺 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
51	47, 50	mpbird 259	. 2 ⊢ (𝜑 → 𝐺 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))
52		estrcco.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
53		estrcco.a	. . . . . . 7 ⊢ 𝐴 = (Base‘𝑋)
54	53	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐴 = (Base‘𝑋))
55	54	eqcomd 2829	. . . . 5 ⊢ (𝜑 → (Base‘𝑋) = 𝐴)
56	55, 42	feq23d 6511	. . . 4 ⊢ (𝜑 → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝐹:𝐴⟶𝐵))
57	52, 56	mpbird 259	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
58		fvexd 6687	. . . 4 ⊢ (𝜑 → (Base‘𝑋) ∈ V)
59	49, 58	elmapd 8422	. . 3 ⊢ (𝜑 → (𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
60	57, 59	mpbird 259	. 2 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))
61		coexg 7636	. . 3 ⊢ ((𝐺 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐺 ∘ 𝐹) ∈ V)
62	51, 60, 61	syl2anc 586	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
63	34, 38, 51, 60, 62	ovmpod 7304	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ⟨cop 4575   × cxp 5555   ∘ ccom 5561  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  1^st c1st 7689  2^nd c2nd 7690   ↑_m cmap 8408  Basecbs 16485  compcco 16579  ExtStrCatcestrc 17374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-hom 16591  df-cco 16592  df-estrc 17375

This theorem is referenced by:  estrccatid  17384  funcestrcsetclem9  17400  funcsetcestrclem9  17415  rngcco  44249  rnghmsubcsetclem2  44254  ringcco  44295  rhmsubcsetclem2  44300