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Theorem estrcco 18185
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 18184 . . 3 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
5 fveq2 6907 . . . . . . 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 6911 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
10 estrcco.x . . . . . . . . 9 (𝜑𝑋𝑈)
11 estrcco.y . . . . . . . . 9 (𝜑𝑌𝑈)
12 op2ndg 8026 . . . . . . . . 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 2775 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
1615fveq2d 6911 . . . . 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 6911 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
1918fveq2d 6911 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st𝑣)) = (Base‘(1st ‘⟨𝑋, 𝑌⟩)))
20 op1stg 8025 . . . . . . . . 9 ((𝑋𝑈𝑌𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2110, 11, 20syl2anc 584 . . . . . . . 8 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2221fveq2d 6911 . . . . . . 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 2775 . . . . 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 2736 . . . 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 5728 . . 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 8103 . . . 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 5878 . . 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 2741 . . . . 5 (𝜑 → (Base‘𝑌) = 𝐵)
43 estrcco.d . . . . . . 7 𝐷 = (Base‘𝑍)
4443a1i 11 . . . . . 6 (𝜑𝐷 = (Base‘𝑍))
4544eqcomd 2741 . . . . 5 (𝜑 → (Base‘𝑍) = 𝐷)
4642, 45feq23d 6732 . . . 4 (𝜑 → (𝐺:(Base‘𝑌)⟶(Base‘𝑍) ↔ 𝐺:𝐵𝐷))
4739, 46mpbird 257 . . 3 (𝜑𝐺:(Base‘𝑌)⟶(Base‘𝑍))
48 fvexd 6922 . . . 4 (𝜑 → (Base‘𝑍) ∈ V)
49 fvexd 6922 . . . 4 (𝜑 → (Base‘𝑌) ∈ V)
5048, 49elmapd 8879 . . 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 2741 . . . . 5 (𝜑 → (Base‘𝑋) = 𝐴)
5655, 42feq23d 6732 . . . 4 (𝜑 → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝐹:𝐴𝐵))
5752, 56mpbird 257 . . 3 (𝜑𝐹:(Base‘𝑋)⟶(Base‘𝑌))
58 fvexd 6922 . . . 4 (𝜑 → (Base‘𝑋) ∈ V)
5949, 58elmapd 8879 . . 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 7952 . . 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 1537  wcel 2106  Vcvv 3478  cop 4637   × cxp 5687  ccom 5693  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  m cmap 8865  Basecbs 17245  compcco 17310  ExtStrCatcestrc 18177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-hom 17322  df-cco 17323  df-estrc 18178
This theorem is referenced by:  estrccatid  18187  funcestrcsetclem9  18204  funcsetcestrclem9  18219  rngcco  20644  rnghmsubcsetclem2  20649  ringcco  20673  rhmsubcsetclem2  20678
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