Theorem estrcco 16770

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 16769	. . 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 6191	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (Base‘𝑧) = (Base‘𝑍))
6	5	adantl 482	. . . . . 6 ⊢ ((𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍) → (Base‘𝑧) = (Base‘𝑍))
7	6	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘𝑧) = (Base‘𝑍))
8		simprl 794	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
9	8	fveq2d 6195	. . . . . . 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 7181	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	10, 11, 12	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
14	13	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
15	9, 14	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
16	15	fveq2d 6195	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(2nd ‘𝑣)) = (Base‘𝑌))
17	7, 16	oveq12d 6668	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))) = ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
18	8	fveq2d 6195	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
19	18	fveq2d 6195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘𝑣)) = (Base‘(1st ‘⟨𝑋, 𝑌⟩)))
20		op1stg 7180	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
21	10, 11, 20	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
22	21	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
23	22	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
24	19, 23	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘𝑣)) = (Base‘𝑋))
25	16, 24	oveq12d 6668	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
26		eqidd 2623	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
27	17, 25, 26	mpt2eq123dv 6717	. . 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		opelxpi 5148	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
29	10, 11, 28	syl2anc 693	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
30		estrcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
31		ovex 6678	. . . . 5 ⊢ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∈ V
32		ovex 6678	. . . . 5 ⊢ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ V
33	31, 32	mpt2ex 7247	. . . 4 ⊢ (𝑔 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↦ (𝑔 ∘ 𝑓)) ∈ V
34	33	a1i 11	. . 3 ⊢ (𝜑 → (𝑔 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↦ (𝑔 ∘ 𝑓)) ∈ V)
35	4, 27, 29, 30, 34	ovmpt2d 6788	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↦ (𝑔 ∘ 𝑓)))
36		simpl 473	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → 𝑔 = 𝐺)
37		simpr 477	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
38	36, 37	coeq12d 5286	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
39	38	adantl 482	. 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
40		estrcco.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷)
41		estrcco.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑌)
42	41	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝑌))
43	42	eqcomd 2628	. . . . 5 ⊢ (𝜑 → (Base‘𝑌) = 𝐵)
44		estrcco.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝑍)
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐷 = (Base‘𝑍))
46	45	eqcomd 2628	. . . . 5 ⊢ (𝜑 → (Base‘𝑍) = 𝐷)
47	43, 46	feq23d 6040	. . . 4 ⊢ (𝜑 → (𝐺:(Base‘𝑌)⟶(Base‘𝑍) ↔ 𝐺:𝐵⟶𝐷))
48	40, 47	mpbird 247	. . 3 ⊢ (𝜑 → 𝐺:(Base‘𝑌)⟶(Base‘𝑍))
49		fvexd 6203	. . . 4 ⊢ (𝜑 → (Base‘𝑍) ∈ V)
50		fvexd 6203	. . . 4 ⊢ (𝜑 → (Base‘𝑌) ∈ V)
51		elmapg 7870	. . . 4 ⊢ (((Base‘𝑍) ∈ V ∧ (Base‘𝑌) ∈ V) → (𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
52	49, 50, 51	syl2anc 693	. . 3 ⊢ (𝜑 → (𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
53	48, 52	mpbird 247	. 2 ⊢ (𝜑 → 𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
54		estrcco.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
55		estrcco.a	. . . . . . 7 ⊢ 𝐴 = (Base‘𝑋)
56	55	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐴 = (Base‘𝑋))
57	56	eqcomd 2628	. . . . 5 ⊢ (𝜑 → (Base‘𝑋) = 𝐴)
58	57, 43	feq23d 6040	. . . 4 ⊢ (𝜑 → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝐹:𝐴⟶𝐵))
59	54, 58	mpbird 247	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
60		fvexd 6203	. . . 4 ⊢ (𝜑 → (Base‘𝑋) ∈ V)
61		elmapg 7870	. . . 4 ⊢ (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
62	50, 60, 61	syl2anc 693	. . 3 ⊢ (𝜑 → (𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
63	59, 62	mpbird 247	. 2 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
64		coexg 7117	. . 3 ⊢ ((𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐺 ∘ 𝐹) ∈ V)
65	53, 63, 64	syl2anc 693	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
66	35, 39, 53, 63, 65	ovmpt2d 6788	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   × cxp 5112   ∘ ccom 5118  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857  Basecbs 15857  compcco 15953  ExtStrCatcestrc 16762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-estrc 16763

This theorem is referenced by:  estrccatid  16772  funcestrcsetclem9  16788  funcsetcestrclem9  16803  rngcco  41971  rnghmsubcsetclem2  41976  ringcco  42017  rhmsubcsetclem2  42022