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Theorem estrcco 16993
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 16992 . . 3 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
5 fveq2 6417 . . . . . . 7 (𝑧 = 𝑍 → (Base‘𝑧) = (Base‘𝑍))
65adantl 469 . . . . . 6 ((𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍) → (Base‘𝑧) = (Base‘𝑍))
76adantl 469 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘𝑧) = (Base‘𝑍))
8 simprl 778 . . . . . . . 8 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
98fveq2d 6421 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
10 estrcco.x . . . . . . . . 9 (𝜑𝑋𝑈)
11 estrcco.y . . . . . . . . 9 (𝜑𝑌𝑈)
12 op2ndg 7420 . . . . . . . . 9 ((𝑋𝑈𝑌𝑈) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1310, 11, 12syl2anc 575 . . . . . . . 8 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1413adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
159, 14eqtrd 2851 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
1615fveq2d 6421 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(2nd𝑣)) = (Base‘𝑌))
177, 16oveq12d 6901 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘𝑧) ↑𝑚 (Base‘(2nd𝑣))) = ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
188fveq2d 6421 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
1918fveq2d 6421 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st𝑣)) = (Base‘(1st ‘⟨𝑋, 𝑌⟩)))
20 op1stg 7419 . . . . . . . . 9 ((𝑋𝑈𝑌𝑈) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2110, 11, 20syl2anc 575 . . . . . . . 8 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2221fveq2d 6421 . . . . . . 7 (𝜑 → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
2322adantr 468 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st ‘⟨𝑋, 𝑌⟩)) = (Base‘𝑋))
2419, 23eqtrd 2851 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (Base‘(1st𝑣)) = (Base‘𝑋))
2516, 24oveq12d 6901 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((Base‘(2nd𝑣)) ↑𝑚 (Base‘(1st𝑣))) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
26 eqidd 2818 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔𝑓) = (𝑔𝑓))
2717, 25, 26mpt2eq123dv 6956 . . 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 5361 . . . 4 ((𝑋𝑈𝑌𝑈) → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
2910, 11, 28syl2anc 575 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑈 × 𝑈))
30 estrcco.z . . 3 (𝜑𝑍𝑈)
31 ovex 6915 . . . . 5 ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∈ V
32 ovex 6915 . . . . 5 ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ∈ V
3331, 32mpt2ex 7489 . . . 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 7027 . 2 (𝜑 → (⟨𝑋, 𝑌· 𝑍) = (𝑔 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)), 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↦ (𝑔𝑓)))
36 simpl 470 . . . 4 ((𝑔 = 𝐺𝑓 = 𝐹) → 𝑔 = 𝐺)
37 simpr 473 . . . 4 ((𝑔 = 𝐺𝑓 = 𝐹) → 𝑓 = 𝐹)
3836, 37coeq12d 5501 . . 3 ((𝑔 = 𝐺𝑓 = 𝐹) → (𝑔𝑓) = (𝐺𝐹))
3938adantl 469 . 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 2823 . . . . 5 (𝜑 → (Base‘𝑌) = 𝐵)
44 estrcco.d . . . . . . 7 𝐷 = (Base‘𝑍)
4544a1i 11 . . . . . 6 (𝜑𝐷 = (Base‘𝑍))
4645eqcomd 2823 . . . . 5 (𝜑 → (Base‘𝑍) = 𝐷)
4743, 46feq23d 6260 . . . 4 (𝜑 → (𝐺:(Base‘𝑌)⟶(Base‘𝑍) ↔ 𝐺:𝐵𝐷))
4840, 47mpbird 248 . . 3 (𝜑𝐺:(Base‘𝑌)⟶(Base‘𝑍))
49 fvexd 6432 . . . 4 (𝜑 → (Base‘𝑍) ∈ V)
50 fvexd 6432 . . . 4 (𝜑 → (Base‘𝑌) ∈ V)
51 elmapg 8114 . . . 4 (((Base‘𝑍) ∈ V ∧ (Base‘𝑌) ∈ V) → (𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
5249, 50, 51syl2anc 575 . . 3 (𝜑 → (𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ↔ 𝐺:(Base‘𝑌)⟶(Base‘𝑍)))
5348, 52mpbird 248 . 2 (𝜑𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)))
54 estrcco.f . . . 4 (𝜑𝐹:𝐴𝐵)
55 estrcco.a . . . . . . 7 𝐴 = (Base‘𝑋)
5655a1i 11 . . . . . 6 (𝜑𝐴 = (Base‘𝑋))
5756eqcomd 2823 . . . . 5 (𝜑 → (Base‘𝑋) = 𝐴)
5857, 43feq23d 6260 . . . 4 (𝜑 → (𝐹:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝐹:𝐴𝐵))
5954, 58mpbird 248 . . 3 (𝜑𝐹:(Base‘𝑋)⟶(Base‘𝑌))
60 fvexd 6432 . . . 4 (𝜑 → (Base‘𝑋) ∈ V)
61 elmapg 8114 . . . 4 (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
6250, 60, 61syl2anc 575 . . 3 (𝜑 → (𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
6359, 62mpbird 248 . 2 (𝜑𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
64 coexg 7356 . . 3 ((𝐺 ∈ ((Base‘𝑍) ↑𝑚 (Base‘𝑌)) ∧ 𝐹 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))) → (𝐺𝐹) ∈ V)
6553, 63, 64syl2anc 575 . 2 (𝜑 → (𝐺𝐹) ∈ V)
6635, 39, 53, 63, 65ovmpt2d 7027 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  Vcvv 3402  cop 4387   × cxp 5322  ccom 5328  wf 6106  cfv 6110  (class class class)co 6883  cmpt2 6885  1st c1st 7405  2nd c2nd 7406  𝑚 cmap 8101  Basecbs 16087  compcco 16184  ExtStrCatcestrc 16985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188  ax-cnex 10286  ax-resscn 10287  ax-1cn 10288  ax-icn 10289  ax-addcl 10290  ax-addrcl 10291  ax-mulcl 10292  ax-mulrcl 10293  ax-mulcom 10294  ax-addass 10295  ax-mulass 10296  ax-distr 10297  ax-i2m1 10298  ax-1ne0 10299  ax-1rid 10300  ax-rnegex 10301  ax-rrecex 10302  ax-cnre 10303  ax-pre-lttri 10304  ax-pre-lttrn 10305  ax-pre-ltadd 10306  ax-pre-mulgt0 10307
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5906  df-ord 5952  df-on 5953  df-lim 5954  df-suc 5955  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-riota 6844  df-ov 6886  df-oprab 6887  df-mpt2 6888  df-om 7305  df-1st 7407  df-2nd 7408  df-wrecs 7651  df-recs 7713  df-rdg 7751  df-1o 7805  df-oadd 7809  df-er 7988  df-map 8103  df-en 8202  df-dom 8203  df-sdom 8204  df-fin 8205  df-pnf 10370  df-mnf 10371  df-xr 10372  df-ltxr 10373  df-le 10374  df-sub 10562  df-neg 10563  df-nn 11315  df-2 11375  df-3 11376  df-4 11377  df-5 11378  df-6 11379  df-7 11380  df-8 11381  df-9 11382  df-n0 11579  df-z 11663  df-dec 11779  df-uz 11924  df-fz 12569  df-struct 16089  df-ndx 16090  df-slot 16091  df-base 16093  df-hom 16196  df-cco 16197  df-estrc 16986
This theorem is referenced by:  estrccatid  16995  funcestrcsetclem9  17012  funcsetcestrclem9  17027  rngcco  42556  rnghmsubcsetclem2  42561  ringcco  42602  rhmsubcsetclem2  42607
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