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Theorem estrcco 18174
Description: Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
Hypotheses
Ref Expression
estrcbas.c 𝐶 = (ExtStrCat‘𝑈)
estrcbas.u (𝜑𝑈𝑉)
estrcco.o · = (comp‘𝐶)
estrcco.x (𝜑𝑋𝑈)
estrcco.y (𝜑𝑌𝑈)
estrcco.z (𝜑𝑍𝑈)
estrcco.a 𝐴 = (Base‘𝑋)
estrcco.b 𝐵 = (Base‘𝑌)
estrcco.d 𝐷 = (Base‘𝑍)
estrcco.f (𝜑𝐹:𝐴𝐵)
estrcco.g (𝜑𝐺:𝐵𝐷)
Assertion
Ref Expression
estrcco (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

Proof of Theorem estrcco
Dummy variables 𝑓 𝑔 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 estrcbas.c . . . 4 𝐶 = (ExtStrCat‘𝑈)
2 estrcbas.u . . . 4 (𝜑𝑈𝑉)
3 estrcco.o . . . 4 · = (comp‘𝐶)
41, 2, 3estrccofval 18173 . . 3 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
5 fveq2 6906 . . . . . . 7 (𝑧 = 𝑍 → (Base‘𝑧) = (Base‘𝑍))
65adantl 481 . . . . . 6 ((𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍) → (Base‘𝑧) = (Base‘𝑍))
76adantl 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⟩)
98fveq2d 6910 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
10 estrcco.x . . . . . . . . 9 (𝜑𝑋𝑈)
11 estrcco.y . . . . . . . . 9 (𝜑𝑌𝑈)
12 op2ndg 8027 . . . . . . . . 9 ((𝑋𝑈𝑌𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1310, 11, 12syl2anc 584 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1413adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
159, 14eqtrd 2777 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
1615fveq2d 6910 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(2nd𝑣)) = (Base‘𝑌))
177, 16oveq12d 7449 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘𝑧) ↑m (Base‘(2nd𝑣))) = ((Base‘𝑍) ↑m (Base‘𝑌)))
188fveq2d 6910 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
1918fveq2d 6910 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st𝑣)) = (Base‘(1st ‘⟨𝑋, 𝑌⟩)))
20 op1stg 8026 . . . . . . . . 9 ((𝑋𝑈𝑌𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2110, 11, 20syl2anc 584 . . . . . . . 8 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2221fveq2d 6910 . . . . . . 7 (𝜑 → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
2322adantr 480 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
2419, 23eqtrd 2777 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st𝑣)) = (Base‘𝑋))
2516, 24oveq12d 7449 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) = ((Base‘𝑌) ↑m (Base‘𝑋)))
26 eqidd 2738 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔𝑓) = (𝑔𝑓))
2717, 25, 26mpoeq123dv 7508 . . 3 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)) = (𝑔 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↦ (𝑔𝑓)))
2810, 11opelxpd 5724 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
29 estrcco.z . . 3 (𝜑𝑍𝑈)
30 ovex 7464 . . . . 5 ((Base‘𝑍) ↑m (Base‘𝑌)) ∈ V
31 ovex 7464 . . . . 5 ((Base‘𝑌) ↑m (Base‘𝑋)) ∈ V
3230, 31mpoex 8104 . . . 4 (𝑔 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↦ (𝑔𝑓)) ∈ V
3332a1i 11 . . 3 (𝜑 → (𝑔 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↦ (𝑔𝑓)) ∈ V)
344, 27, 28, 29, 33ovmpod 7585 . 2 (𝜑 → (⟨𝑋, 𝑌· 𝑍) = (𝑔 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↦ (𝑔𝑓)))
35 simpl 482 . . . 4 ((𝑔 = 𝐺𝑓 = 𝐹) → 𝑔 = 𝐺)
36 simpr 484 . . . 4 ((𝑔 = 𝐺𝑓 = 𝐹) → 𝑓 = 𝐹)
3735, 36coeq12d 5875 . . 3 ((𝑔 = 𝐺𝑓 = 𝐹) → (𝑔𝑓) = (𝐺𝐹))
3837adantl 481 . 2 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑔𝑓) = (𝐺𝐹))
39 estrcco.g . . . 4 (𝜑𝐺:𝐵𝐷)
40 estrcco.b . . . . . . 7 𝐵 = (Base‘𝑌)
4140a1i 11 . . . . . 6 (𝜑𝐵 = (Base‘𝑌))
4241eqcomd 2743 . . . . 5 (𝜑 → (Base‘𝑌) = 𝐵)
43 estrcco.d . . . . . . 7 𝐷 = (Base‘𝑍)
4443a1i 11 . . . . . 6 (𝜑𝐷 = (Base‘𝑍))
4544eqcomd 2743 . . . . 5 (𝜑 → (Base‘𝑍) = 𝐷)
4642, 45feq23d 6731 . . . 4 (𝜑 → (𝐺:(Base‘𝑌)⟶(Base‘𝑍) ↔ 𝐺:𝐵𝐷))
4739, 46mpbird 257 . . 3 (𝜑𝐺:(Base‘𝑌)⟶(Base‘𝑍))
48 fvexd 6921 . . . 4 (𝜑 → (Base‘𝑍) ∈ V)
49 fvexd 6921 . . . 4 (𝜑 → (Base‘𝑌) ∈ V)
5048, 49elmapd 8880 . . 3 (𝜑 → (𝐺 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
5147, 50mpbird 257 . 2 (𝜑𝐺 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)))
52 estrcco.f . . . 4 (𝜑𝐹:𝐴𝐵)
53 estrcco.a . . . . . . 7 𝐴 = (Base‘𝑋)
5453a1i 11 . . . . . 6 (𝜑𝐴 = (Base‘𝑋))
5554eqcomd 2743 . . . . 5 (𝜑 → (Base‘𝑋) = 𝐴)
5655, 42feq23d 6731 . . . 4 (𝜑 → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝐹:𝐴𝐵))
5752, 56mpbird 257 . . 3 (𝜑𝐹:(Base‘𝑋)⟶(Base‘𝑌))
58 fvexd 6921 . . . 4 (𝜑 → (Base‘𝑋) ∈ V)
5949, 58elmapd 8880 . . 3 (𝜑 → (𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
6057, 59mpbird 257 . 2 (𝜑𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))
61 coexg 7951 . . 3 ((𝐺 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐺𝐹) ∈ V)
6251, 60, 61syl2anc 584 . 2 (𝜑 → (𝐺𝐹) ∈ V)
6334, 38, 51, 60, 62ovmpod 7585 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632   × cxp 5683  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  m cmap 8866  Basecbs 17247  compcco 17309  ExtStrCatcestrc 18166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-estrc 18167
This theorem is referenced by:  estrccatid  18176  funcestrcsetclem9  18193  funcsetcestrclem9  18208  rngcco  20627  rnghmsubcsetclem2  20632  ringcco  20656  rhmsubcsetclem2  20661
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