Theorem estrcco 18067

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 18066	. . 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 6844	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (Base‘𝑧) = (Base‘𝑍))
6	5	adantl 481	. . . . . 6 ⊢ ((𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍) → (Base‘𝑧) = (Base‘𝑍))
7	6	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘𝑧) = (Base‘𝑍))
8		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
9	8	fveq2d 6848	. . . . . . 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 7958	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	10, 11, 12	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
14	13	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
15	9, 14	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
16	15	fveq2d 6848	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(2nd ‘𝑣)) = (Base‘𝑌))
17	7, 16	oveq12d 7388	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘𝑧) ↑m (Base‘(2nd ‘𝑣))) = ((Base‘𝑍) ↑m (Base‘𝑌)))
18	8	fveq2d 6848	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
19	18	fveq2d 6848	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘𝑣)) = (Base‘(1st ‘⟨𝑋, 𝑌⟩)))
20		op1stg 7957	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
21	10, 11, 20	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
22	21	fveq2d 6848	. . . . . . 7 ⊢ (𝜑 → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
23	22	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
24	19, 23	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘𝑣)) = (Base‘𝑋))
25	16, 24	oveq12d 7388	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘(2nd ‘𝑣)) ↑m (Base‘(1st ‘𝑣))) = ((Base‘𝑌) ↑m (Base‘𝑋)))
26		eqidd 2738	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
27	17, 25, 26	mpoeq123dv 7445	. . 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 5673	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
29		estrcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
30		ovex 7403	. . . . 5 ⊢ ((Base‘𝑍) ↑m (Base‘𝑌)) ∈ V
31		ovex 7403	. . . . 5 ⊢ ((Base‘𝑌) ↑m (Base‘𝑋)) ∈ V
32	30, 31	mpoex 8035	. . . 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 7522	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↦ (𝑔 ∘ 𝑓)))
35		simpl 482	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → 𝑔 = 𝐺)
36		simpr 484	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
37	35, 36	coeq12d 5823	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
38	37	adantl 481	. 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 2743	. . . . 5 ⊢ (𝜑 → (Base‘𝑌) = 𝐵)
43		estrcco.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝑍)
44	43	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐷 = (Base‘𝑍))
45	44	eqcomd 2743	. . . . 5 ⊢ (𝜑 → (Base‘𝑍) = 𝐷)
46	42, 45	feq23d 6667	. . . 4 ⊢ (𝜑 → (𝐺:(Base‘𝑌)⟶(Base‘𝑍) ↔ 𝐺:𝐵⟶𝐷))
47	39, 46	mpbird 257	. . 3 ⊢ (𝜑 → 𝐺:(Base‘𝑌)⟶(Base‘𝑍))
48		fvexd 6859	. . . 4 ⊢ (𝜑 → (Base‘𝑍) ∈ V)
49		fvexd 6859	. . . 4 ⊢ (𝜑 → (Base‘𝑌) ∈ V)
50	48, 49	elmapd 8791	. . 3 ⊢ (𝜑 → (𝐺 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
51	47, 50	mpbird 257	. 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 2743	. . . . 5 ⊢ (𝜑 → (Base‘𝑋) = 𝐴)
56	55, 42	feq23d 6667	. . . 4 ⊢ (𝜑 → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝐹:𝐴⟶𝐵))
57	52, 56	mpbird 257	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
58		fvexd 6859	. . . 4 ⊢ (𝜑 → (Base‘𝑋) ∈ V)
59	49, 58	elmapd 8791	. . 3 ⊢ (𝜑 → (𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
60	57, 59	mpbird 257	. 2 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))
61		coexg 7883	. . 3 ⊢ ((𝐺 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐺 ∘ 𝐹) ∈ V)
62	51, 60, 61	syl2anc 585	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
63	34, 38, 51, 60, 62	ovmpod 7522	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ⟨cop 4588   × cxp 5632   ∘ ccom 5638  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372  1^st c1st 7943  2^nd c2nd 7944   ↑_m cmap 8777  Basecbs 17150  compcco 17203  ExtStrCatcestrc 18059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-map 8779  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-z 12503  df-dec 12622  df-uz 12766  df-fz 13438  df-struct 17088  df-slot 17123  df-ndx 17135  df-base 17151  df-hom 17215  df-cco 17216  df-estrc 18060

This theorem is referenced by:  estrccatid  18069  funcestrcsetclem9  18085  funcsetcestrclem9  18100  rngcco  20577  rnghmsubcsetclem2  20582  ringcco  20606  rhmsubcsetclem2  20611