Theorem estrcco 16593

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 16592	. . 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 6103	. . . . . . 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 790	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
9	8	fveq2d 6107	. . . . . . 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 7072	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
13	10, 11, 12	syl2anc 691	. . . . . . . 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 2644	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑣) = 𝑌)
16	15	fveq2d 6107	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(2nd ‘𝑣)) = (Base‘𝑌))
17	7, 16	oveq12d 6567	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))) = ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
18	8	fveq2d 6107	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
19	18	fveq2d 6107	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘𝑣)) = (Base‘(1st ‘⟨𝑋, 𝑌⟩)))
20		op1stg 7071	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
21	10, 11, 20	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
22	21	fveq2d 6107	. . . . . . 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 2644	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘𝑣)) = (Base‘𝑋))
25	16, 24	oveq12d 6567	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
26		eqidd 2611	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
27	17, 25, 26	mpt2eq123dv 6615	. . 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 5072	. . . 4 ⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
29	10, 11, 28	syl2anc 691	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
30		estrcco.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
31		ovex 6577	. . . . 5 ⊢ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∈ V
32		ovex 6577	. . . . 5 ⊢ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ V
33	31, 32	mpt2ex 7136	. . . 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 6686	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (𝑔 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↦ (𝑔 ∘ 𝑓)))
36		simpl 472	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → 𝑔 = 𝐺)
37		simpr 476	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
38	36, 37	coeq12d 5208	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
39	38	adantl 481	. 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 2616	. . . . 5 ⊢ (𝜑 → (Base‘𝑌) = 𝐵)
44		estrcco.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝑍)
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐷 = (Base‘𝑍))
46	45	eqcomd 2616	. . . . 5 ⊢ (𝜑 → (Base‘𝑍) = 𝐷)
47	43, 46	feq23d 5953	. . . 4 ⊢ (𝜑 → (𝐺:(Base‘𝑌)⟶(Base‘𝑍) ↔ 𝐺:𝐵⟶𝐷))
48	40, 47	mpbird 246	. . 3 ⊢ (𝜑 → 𝐺:(Base‘𝑌)⟶(Base‘𝑍))
49		fvex 6113	. . . . 5 ⊢ (Base‘𝑍) ∈ V
50	49	a1i 11	. . . 4 ⊢ (𝜑 → (Base‘𝑍) ∈ V)
51		fvex 6113	. . . . 5 ⊢ (Base‘𝑌) ∈ V
52	51	a1i 11	. . . 4 ⊢ (𝜑 → (Base‘𝑌) ∈ V)
53		elmapg 7757	. . . 4 ⊢ (((Base‘𝑍) ∈ V ∧ (Base‘𝑌) ∈ V) → (𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
54	50, 52, 53	syl2anc 691	. . 3 ⊢ (𝜑 → (𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
55	48, 54	mpbird 246	. 2 ⊢ (𝜑 → 𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
56		estrcco.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
57		estrcco.a	. . . . . . 7 ⊢ 𝐴 = (Base‘𝑋)
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐴 = (Base‘𝑋))
59	58	eqcomd 2616	. . . . 5 ⊢ (𝜑 → (Base‘𝑋) = 𝐴)
60	59, 43	feq23d 5953	. . . 4 ⊢ (𝜑 → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝐹:𝐴⟶𝐵))
61	56, 60	mpbird 246	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
62		fvex 6113	. . . . 5 ⊢ (Base‘𝑋) ∈ V
63	62	a1i 11	. . . 4 ⊢ (𝜑 → (Base‘𝑋) ∈ V)
64		elmapg 7757	. . . 4 ⊢ (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
65	52, 63, 64	syl2anc 691	. . 3 ⊢ (𝜑 → (𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
66	61, 65	mpbird 246	. 2 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
67		coexg 7010	. . 3 ⊢ ((𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐺 ∘ 𝐹) ∈ V)
68	55, 66, 67	syl2anc 691	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
69	35, 39, 55, 66, 68	ovmpt2d 6686	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  Basecbs 15695  compcco 15780  ExtStrCatcestrc 16585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-estrc 16586

This theorem is referenced by:  estrccatid  16595  funcestrcsetclem9  16611  funcsetcestrclem9  16626  rngcco  41763  rnghmsubcsetclem2  41768  ringcco  41809  rhmsubcsetclem2  41814