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Theorem estrcco 18123
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 18122 . . 3 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
5 fveq2 6896 . . . . . . 7 (𝑧 = 𝑍 → (Base‘𝑧) = (Base‘𝑍))
65adantl 480 . . . . . 6 ((𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍) → (Base‘𝑧) = (Base‘𝑍))
76adantl 480 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘𝑧) = (Base‘𝑍))
8 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
98fveq2d 6900 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
10 estrcco.x . . . . . . . . 9 (𝜑𝑋𝑈)
11 estrcco.y . . . . . . . . 9 (𝜑𝑌𝑈)
12 op2ndg 8007 . . . . . . . . 9 ((𝑋𝑈𝑌𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1310, 11, 12syl2anc 582 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1413adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
159, 14eqtrd 2765 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
1615fveq2d 6900 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(2nd𝑣)) = (Base‘𝑌))
177, 16oveq12d 7437 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘𝑧) ↑m (Base‘(2nd𝑣))) = ((Base‘𝑍) ↑m (Base‘𝑌)))
188fveq2d 6900 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
1918fveq2d 6900 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st𝑣)) = (Base‘(1st ‘⟨𝑋, 𝑌⟩)))
20 op1stg 8006 . . . . . . . . 9 ((𝑋𝑈𝑌𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2110, 11, 20syl2anc 582 . . . . . . . 8 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2221fveq2d 6900 . . . . . . 7 (𝜑 → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
2322adantr 479 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
2419, 23eqtrd 2765 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st𝑣)) = (Base‘𝑋))
2516, 24oveq12d 7437 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) = ((Base‘𝑌) ↑m (Base‘𝑋)))
26 eqidd 2726 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔𝑓) = (𝑔𝑓))
2717, 25, 26mpoeq123dv 7495 . . 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 5717 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
29 estrcco.z . . 3 (𝜑𝑍𝑈)
30 ovex 7452 . . . . 5 ((Base‘𝑍) ↑m (Base‘𝑌)) ∈ V
31 ovex 7452 . . . . 5 ((Base‘𝑌) ↑m (Base‘𝑋)) ∈ V
3230, 31mpoex 8084 . . . 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 7573 . 2 (𝜑 → (⟨𝑋, 𝑌· 𝑍) = (𝑔 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↦ (𝑔𝑓)))
35 simpl 481 . . . 4 ((𝑔 = 𝐺𝑓 = 𝐹) → 𝑔 = 𝐺)
36 simpr 483 . . . 4 ((𝑔 = 𝐺𝑓 = 𝐹) → 𝑓 = 𝐹)
3735, 36coeq12d 5867 . . 3 ((𝑔 = 𝐺𝑓 = 𝐹) → (𝑔𝑓) = (𝐺𝐹))
3837adantl 480 . 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 2731 . . . . 5 (𝜑 → (Base‘𝑌) = 𝐵)
43 estrcco.d . . . . . . 7 𝐷 = (Base‘𝑍)
4443a1i 11 . . . . . 6 (𝜑𝐷 = (Base‘𝑍))
4544eqcomd 2731 . . . . 5 (𝜑 → (Base‘𝑍) = 𝐷)
4642, 45feq23d 6718 . . . 4 (𝜑 → (𝐺:(Base‘𝑌)⟶(Base‘𝑍) ↔ 𝐺:𝐵𝐷))
4739, 46mpbird 256 . . 3 (𝜑𝐺:(Base‘𝑌)⟶(Base‘𝑍))
48 fvexd 6911 . . . 4 (𝜑 → (Base‘𝑍) ∈ V)
49 fvexd 6911 . . . 4 (𝜑 → (Base‘𝑌) ∈ V)
5048, 49elmapd 8859 . . 3 (𝜑 → (𝐺 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
5147, 50mpbird 256 . 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 2731 . . . . 5 (𝜑 → (Base‘𝑋) = 𝐴)
5655, 42feq23d 6718 . . . 4 (𝜑 → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝐹:𝐴𝐵))
5752, 56mpbird 256 . . 3 (𝜑𝐹:(Base‘𝑋)⟶(Base‘𝑌))
58 fvexd 6911 . . . 4 (𝜑 → (Base‘𝑋) ∈ V)
5949, 58elmapd 8859 . . 3 (𝜑 → (𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
6057, 59mpbird 256 . 2 (𝜑𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))
61 coexg 7937 . . 3 ((𝐺 ∈ ((Base‘𝑍) ↑m (Base‘𝑌)) ∧ 𝐹 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))) → (𝐺𝐹) ∈ V)
6251, 60, 61syl2anc 582 . 2 (𝜑 → (𝐺𝐹) ∈ V)
6334, 38, 51, 60, 62ovmpod 7573 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  cop 4636   × cxp 5676  ccom 5682  wf 6545  cfv 6549  (class class class)co 7419  cmpo 7421  1st c1st 7992  2nd c2nd 7993  m cmap 8845  Basecbs 17183  compcco 17248  ExtStrCatcestrc 18115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-fz 13520  df-struct 17119  df-slot 17154  df-ndx 17166  df-base 17184  df-hom 17260  df-cco 17261  df-estrc 18116
This theorem is referenced by:  estrccatid  18125  funcestrcsetclem9  18142  funcsetcestrclem9  18157  rngcco  20572  rnghmsubcsetclem2  20577  ringcco  20601  rhmsubcsetclem2  20606
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