Theorem estrcco 16993

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 16992	. . 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 6417	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (Base‘𝑧) = (Base‘𝑍))
6	5	adantl 469	. . . . . 6 ⊢ ((𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍) → (Base‘𝑧) = (Base‘𝑍))
7	6	adantl 469	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘𝑧) = (Base‘𝑍))
8		simprl 778	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
9	8	fveq2d 6421	. . . . . . 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 7420	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	10, 11, 12	syl2anc 575	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
14	13	adantr 468	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
15	9, 14	eqtrd 2851	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
16	15	fveq2d 6421	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(2nd ‘𝑣)) = (Base‘𝑌))
17	7, 16	oveq12d 6901	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))) = ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
18	8	fveq2d 6421	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
19	18	fveq2d 6421	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘𝑣)) = (Base‘(1st ‘⟨𝑋, 𝑌⟩)))
20		op1stg 7419	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
21	10, 11, 20	syl2anc 575	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
22	21	fveq2d 6421	. . . . . . 7 ⊢ (𝜑 → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
23	22	adantr 468	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
24	19, 23	eqtrd 2851	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘𝑣)) = (Base‘𝑋))
25	16, 24	oveq12d 6901	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
26		eqidd 2818	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
27	17, 25, 26	mpt2eq123dv 6956	. . 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 5361	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
29	10, 11, 28	syl2anc 575	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
30		estrcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
31		ovex 6915	. . . . 5 ⊢ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∈ V
32		ovex 6915	. . . . 5 ⊢ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ V
33	31, 32	mpt2ex 7489	. . . 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 7027	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↦ (𝑔 ∘ 𝑓)))
36		simpl 470	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → 𝑔 = 𝐺)
37		simpr 473	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
38	36, 37	coeq12d 5501	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
39	38	adantl 469	. 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 2823	. . . . 5 ⊢ (𝜑 → (Base‘𝑌) = 𝐵)
44		estrcco.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝑍)
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐷 = (Base‘𝑍))
46	45	eqcomd 2823	. . . . 5 ⊢ (𝜑 → (Base‘𝑍) = 𝐷)
47	43, 46	feq23d 6260	. . . 4 ⊢ (𝜑 → (𝐺:(Base‘𝑌)⟶(Base‘𝑍) ↔ 𝐺:𝐵⟶𝐷))
48	40, 47	mpbird 248	. . 3 ⊢ (𝜑 → 𝐺:(Base‘𝑌)⟶(Base‘𝑍))
49		fvexd 6432	. . . 4 ⊢ (𝜑 → (Base‘𝑍) ∈ V)
50		fvexd 6432	. . . 4 ⊢ (𝜑 → (Base‘𝑌) ∈ V)
51		elmapg 8114	. . . 4 ⊢ (((Base‘𝑍) ∈ V ∧ (Base‘𝑌) ∈ V) → (𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
52	49, 50, 51	syl2anc 575	. . 3 ⊢ (𝜑 → (𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
53	48, 52	mpbird 248	. 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 2823	. . . . 5 ⊢ (𝜑 → (Base‘𝑋) = 𝐴)
58	57, 43	feq23d 6260	. . . 4 ⊢ (𝜑 → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝐹:𝐴⟶𝐵))
59	54, 58	mpbird 248	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
60		fvexd 6432	. . . 4 ⊢ (𝜑 → (Base‘𝑋) ∈ V)
61		elmapg 8114	. . . 4 ⊢ (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
62	50, 60, 61	syl2anc 575	. . 3 ⊢ (𝜑 → (𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
63	59, 62	mpbird 248	. 2 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
64		coexg 7356	. . 3 ⊢ ((𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐺 ∘ 𝐹) ∈ V)
65	53, 63, 64	syl2anc 575	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
66	35, 39, 53, 63, 65	ovmpt2d 7027	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   = wceq 1637   ∈ wcel 2157  Vcvv 3402  ⟨cop 4387   × cxp 5322   ∘ ccom 5328  ⟶wf 6106  ‘cfv 6110  (class class class)co 6883   ↦ cmpt2 6885  1^st c1st 7405  2^nd c2nd 7406   ↑_𝑚 cmap 8101  Basecbs 16087  compcco 16184  ExtStrCatcestrc 16985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188  ax-cnex 10286  ax-resscn 10287  ax-1cn 10288  ax-icn 10289  ax-addcl 10290  ax-addrcl 10291  ax-mulcl 10292  ax-mulrcl 10293  ax-mulcom 10294  ax-addass 10295  ax-mulass 10296  ax-distr 10297  ax-i2m1 10298  ax-1ne0 10299  ax-1rid 10300  ax-rnegex 10301  ax-rrecex 10302  ax-cnre 10303  ax-pre-lttri 10304  ax-pre-lttrn 10305  ax-pre-ltadd 10306  ax-pre-mulgt0 10307

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5906  df-ord 5952  df-on 5953  df-lim 5954  df-suc 5955  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-riota 6844  df-ov 6886  df-oprab 6887  df-mpt2 6888  df-om 7305  df-1st 7407  df-2nd 7408  df-wrecs 7651  df-recs 7713  df-rdg 7751  df-1o 7805  df-oadd 7809  df-er 7988  df-map 8103  df-en 8202  df-dom 8203  df-sdom 8204  df-fin 8205  df-pnf 10370  df-mnf 10371  df-xr 10372  df-ltxr 10373  df-le 10374  df-sub 10562  df-neg 10563  df-nn 11315  df-2 11375  df-3 11376  df-4 11377  df-5 11378  df-6 11379  df-7 11380  df-8 11381  df-9 11382  df-n0 11579  df-z 11663  df-dec 11779  df-uz 11924  df-fz 12569  df-struct 16089  df-ndx 16090  df-slot 16091  df-base 16093  df-hom 16196  df-cco 16197  df-estrc 16986

This theorem is referenced by:  estrccatid  16995  funcestrcsetclem9  17012  funcsetcestrclem9  17027  rngcco  42556  rnghmsubcsetclem2  42561  ringcco  42602  rhmsubcsetclem2  42607