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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 (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 16592 . . 3 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
5 fveq2 6103 . . . . . . 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 790 . . . . . . . 8 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
98fveq2d 6107 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
10 estrcco.x . . . . . . . . 9 (𝜑𝑋𝑈)
11 estrcco.y . . . . . . . . 9 (𝜑𝑌𝑈)
12 op2ndg 7072 . . . . . . . . 9 ((𝑋𝑈𝑌𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1310, 11, 12syl2anc 691 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1413adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
159, 14eqtrd 2644 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
1615fveq2d 6107 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(2nd𝑣)) = (Base‘𝑌))
177, 16oveq12d 6567 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))) = ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
188fveq2d 6107 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
1918fveq2d 6107 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st𝑣)) = (Base‘(1st ‘⟨𝑋, 𝑌⟩)))
20 op1stg 7071 . . . . . . . . 9 ((𝑋𝑈𝑌𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2110, 11, 20syl2anc 691 . . . . . . . 8 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2221fveq2d 6107 . . . . . . 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 2644 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st𝑣)) = (Base‘𝑋))
2516, 24oveq12d 6567 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
26 eqidd 2611 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔𝑓) = (𝑔𝑓))
2717, 25, 26mpt2eq123dv 6615 . . 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)))
28 opelxpi 5072 . . . 4 ((𝑋𝑈𝑌𝑈) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
2910, 11, 28syl2anc 691 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
30 estrcco.z . . 3 (𝜑𝑍𝑈)
31 ovex 6577 . . . . 5 ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∈ V
32 ovex 6577 . . . . 5 ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ V
3331, 32mpt2ex 7136 . . . 4 (𝑔 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↦ (𝑔𝑓)) ∈ V
3433a1i 11 . . 3 (𝜑 → (𝑔 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↦ (𝑔𝑓)) ∈ V)
354, 27, 29, 30, 34ovmpt2d 6686 . 2 (𝜑 → (⟨𝑋, 𝑌· 𝑍) = (𝑔 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↦ (𝑔𝑓)))
36 simpl 472 . . . 4 ((𝑔 = 𝐺𝑓 = 𝐹) → 𝑔 = 𝐺)
37 simpr 476 . . . 4 ((𝑔 = 𝐺𝑓 = 𝐹) → 𝑓 = 𝐹)
3836, 37coeq12d 5208 . . 3 ((𝑔 = 𝐺𝑓 = 𝐹) → (𝑔𝑓) = (𝐺𝐹))
3938adantl 481 . 2 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑔𝑓) = (𝐺𝐹))
40 estrcco.g . . . 4 (𝜑𝐺:𝐵𝐷)
41 estrcco.b . . . . . . 7 𝐵 = (Base‘𝑌)
4241a1i 11 . . . . . 6 (𝜑𝐵 = (Base‘𝑌))
4342eqcomd 2616 . . . . 5 (𝜑 → (Base‘𝑌) = 𝐵)
44 estrcco.d . . . . . . 7 𝐷 = (Base‘𝑍)
4544a1i 11 . . . . . 6 (𝜑𝐷 = (Base‘𝑍))
4645eqcomd 2616 . . . . 5 (𝜑 → (Base‘𝑍) = 𝐷)
4743, 46feq23d 5953 . . . 4 (𝜑 → (𝐺:(Base‘𝑌)⟶(Base‘𝑍) ↔ 𝐺:𝐵𝐷))
4840, 47mpbird 246 . . 3 (𝜑𝐺:(Base‘𝑌)⟶(Base‘𝑍))
49 fvex 6113 . . . . 5 (Base‘𝑍) ∈ V
5049a1i 11 . . . 4 (𝜑 → (Base‘𝑍) ∈ V)
51 fvex 6113 . . . . 5 (Base‘𝑌) ∈ V
5251a1i 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)))
5450, 52, 53syl2anc 691 . . 3 (𝜑 → (𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
5548, 54mpbird 246 . 2 (𝜑𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
56 estrcco.f . . . 4 (𝜑𝐹:𝐴𝐵)
57 estrcco.a . . . . . . 7 𝐴 = (Base‘𝑋)
5857a1i 11 . . . . . 6 (𝜑𝐴 = (Base‘𝑋))
5958eqcomd 2616 . . . . 5 (𝜑 → (Base‘𝑋) = 𝐴)
6059, 43feq23d 5953 . . . 4 (𝜑 → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝐹:𝐴𝐵))
6156, 60mpbird 246 . . 3 (𝜑𝐹:(Base‘𝑋)⟶(Base‘𝑌))
62 fvex 6113 . . . . 5 (Base‘𝑋) ∈ V
6362a1i 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)))
6552, 63, 64syl2anc 691 . . 3 (𝜑 → (𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
6661, 65mpbird 246 . 2 (𝜑𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
67 coexg 7010 . . 3 ((𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐺𝐹) ∈ V)
6855, 66, 67syl2anc 691 . 2 (𝜑 → (𝐺𝐹) ∈ V)
6935, 39, 55, 66, 68ovmpt2d 6686 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
 Colors of variables: wff setvar class 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  1st c1st 7057  2nd 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
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