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Theorem estrcco 18090
Description: Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
Hypotheses
Ref Expression
estrcbas.c 𝐢 = (ExtStrCatβ€˜π‘ˆ)
estrcbas.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
estrcco.o Β· = (compβ€˜πΆ)
estrcco.x (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
estrcco.y (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
estrcco.z (πœ‘ β†’ 𝑍 ∈ π‘ˆ)
estrcco.a 𝐴 = (Baseβ€˜π‘‹)
estrcco.b 𝐡 = (Baseβ€˜π‘Œ)
estrcco.d 𝐷 = (Baseβ€˜π‘)
estrcco.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
estrcco.g (πœ‘ β†’ 𝐺:𝐡⟢𝐷)
Assertion
Ref Expression
estrcco (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) = (𝐺 ∘ 𝐹))

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 𝐢 = (ExtStrCatβ€˜π‘ˆ)
2 estrcbas.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
3 estrcco.o . . . 4 Β· = (compβ€˜πΆ)
41, 2, 3estrccofval 18089 . . 3 (πœ‘ β†’ Β· = (𝑣 ∈ (π‘ˆ Γ— π‘ˆ), 𝑧 ∈ π‘ˆ ↦ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓))))
5 fveq2 6884 . . . . . . 7 (𝑧 = 𝑍 β†’ (Baseβ€˜π‘§) = (Baseβ€˜π‘))
65adantl 481 . . . . . 6 ((𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍) β†’ (Baseβ€˜π‘§) = (Baseβ€˜π‘))
76adantl 481 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (Baseβ€˜π‘§) = (Baseβ€˜π‘))
8 simprl 768 . . . . . . . 8 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑣 = βŸ¨π‘‹, π‘ŒβŸ©)
98fveq2d 6888 . . . . . . 7 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘£) = (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©))
10 estrcco.x . . . . . . . . 9 (πœ‘ β†’ 𝑋 ∈ π‘ˆ)
11 estrcco.y . . . . . . . . 9 (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
12 op2ndg 7984 . . . . . . . . 9 ((𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
1310, 11, 12syl2anc 583 . . . . . . . 8 (πœ‘ β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
1413adantr 480 . . . . . . 7 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
159, 14eqtrd 2766 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘£) = π‘Œ)
1615fveq2d 6888 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (Baseβ€˜(2nd β€˜π‘£)) = (Baseβ€˜π‘Œ))
177, 16oveq12d 7422 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))) = ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Œ)))
188fveq2d 6888 . . . . . . 7 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜π‘£) = (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©))
1918fveq2d 6888 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (Baseβ€˜(1st β€˜π‘£)) = (Baseβ€˜(1st β€˜βŸ¨π‘‹, π‘ŒβŸ©)))
20 op1stg 7983 . . . . . . . . 9 ((𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ) β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
2110, 11, 20syl2anc 583 . . . . . . . 8 (πœ‘ β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
2221fveq2d 6888 . . . . . . 7 (πœ‘ β†’ (Baseβ€˜(1st β€˜βŸ¨π‘‹, π‘ŒβŸ©)) = (Baseβ€˜π‘‹))
2322adantr 480 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (Baseβ€˜(1st β€˜βŸ¨π‘‹, π‘ŒβŸ©)) = (Baseβ€˜π‘‹))
2419, 23eqtrd 2766 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (Baseβ€˜(1st β€˜π‘£)) = (Baseβ€˜π‘‹))
2516, 24oveq12d 7422 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) = ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)))
26 eqidd 2727 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
2717, 25, 26mpoeq123dv 7479 . . 3 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Œ)), 𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↦ (𝑔 ∘ 𝑓)))
2810, 11opelxpd 5708 . . 3 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (π‘ˆ Γ— π‘ˆ))
29 estrcco.z . . 3 (πœ‘ β†’ 𝑍 ∈ π‘ˆ)
30 ovex 7437 . . . . 5 ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Œ)) ∈ V
31 ovex 7437 . . . . 5 ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ∈ V
3230, 31mpoex 8062 . . . 4 (𝑔 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Œ)), 𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↦ (𝑔 ∘ 𝑓)) ∈ V
3332a1i 11 . . 3 (πœ‘ β†’ (𝑔 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Œ)), 𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↦ (𝑔 ∘ 𝑓)) ∈ V)
344, 27, 28, 29, 33ovmpod 7555 . 2 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍) = (𝑔 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Œ)), 𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↦ (𝑔 ∘ 𝑓)))
35 simpl 482 . . . 4 ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) β†’ 𝑔 = 𝐺)
36 simpr 484 . . . 4 ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
3735, 36coeq12d 5857 . . 3 ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) β†’ (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
3837adantl 481 . 2 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
39 estrcco.g . . . 4 (πœ‘ β†’ 𝐺:𝐡⟢𝐷)
40 estrcco.b . . . . . . 7 𝐡 = (Baseβ€˜π‘Œ)
4140a1i 11 . . . . . 6 (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘Œ))
4241eqcomd 2732 . . . . 5 (πœ‘ β†’ (Baseβ€˜π‘Œ) = 𝐡)
43 estrcco.d . . . . . . 7 𝐷 = (Baseβ€˜π‘)
4443a1i 11 . . . . . 6 (πœ‘ β†’ 𝐷 = (Baseβ€˜π‘))
4544eqcomd 2732 . . . . 5 (πœ‘ β†’ (Baseβ€˜π‘) = 𝐷)
4642, 45feq23d 6705 . . . 4 (πœ‘ β†’ (𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘) ↔ 𝐺:𝐡⟢𝐷))
4739, 46mpbird 257 . . 3 (πœ‘ β†’ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘))
48 fvexd 6899 . . . 4 (πœ‘ β†’ (Baseβ€˜π‘) ∈ V)
49 fvexd 6899 . . . 4 (πœ‘ β†’ (Baseβ€˜π‘Œ) ∈ V)
5048, 49elmapd 8833 . . 3 (πœ‘ β†’ (𝐺 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Œ)) ↔ 𝐺:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘)))
5147, 50mpbird 257 . 2 (πœ‘ β†’ 𝐺 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Œ)))
52 estrcco.f . . . 4 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
53 estrcco.a . . . . . . 7 𝐴 = (Baseβ€˜π‘‹)
5453a1i 11 . . . . . 6 (πœ‘ β†’ 𝐴 = (Baseβ€˜π‘‹))
5554eqcomd 2732 . . . . 5 (πœ‘ β†’ (Baseβ€˜π‘‹) = 𝐴)
5655, 42feq23d 6705 . . . 4 (πœ‘ β†’ (𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ↔ 𝐹:𝐴⟢𝐡))
5752, 56mpbird 257 . . 3 (πœ‘ β†’ 𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
58 fvexd 6899 . . . 4 (πœ‘ β†’ (Baseβ€˜π‘‹) ∈ V)
5949, 58elmapd 8833 . . 3 (πœ‘ β†’ (𝐹 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↔ 𝐹:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ)))
6057, 59mpbird 257 . 2 (πœ‘ β†’ 𝐹 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)))
61 coexg 7916 . . 3 ((𝐺 ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Œ)) ∧ 𝐹 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹))) β†’ (𝐺 ∘ 𝐹) ∈ V)
6251, 60, 61syl2anc 583 . 2 (πœ‘ β†’ (𝐺 ∘ 𝐹) ∈ V)
6334, 38, 51, 60, 62ovmpod 7555 1 (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) = (𝐺 ∘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  βŸ¨cop 4629   Γ— cxp 5667   ∘ ccom 5673  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  1st c1st 7969  2nd c2nd 7970   ↑m cmap 8819  Basecbs 17150  compcco 17215  ExtStrCatcestrc 18082
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-z 12560  df-dec 12679  df-uz 12824  df-fz 13488  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17151  df-hom 17227  df-cco 17228  df-estrc 18083
This theorem is referenced by:  estrccatid  18092  funcestrcsetclem9  18109  funcsetcestrclem9  18124  rngcco  20520  rnghmsubcsetclem2  20525  ringcco  20549  rhmsubcsetclem2  20554
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