Theorem xpcco2nd 18249

Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
xpcco1st.t	⊢ 𝑇 = (𝐶 ×c 𝐷)
xpcco1st.b	⊢ 𝐵 = (Base‘𝑇)
xpcco1st.k	⊢ 𝐾 = (Hom ‘𝑇)
xpcco1st.o	⊢ 𝑂 = (comp‘𝑇)
xpcco1st.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
xpcco1st.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
xpcco1st.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
xpcco1st.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))
xpcco1st.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))
xpcco2nd.1	⊢ · = (comp‘𝐷)

Assertion

Ref	Expression
xpcco2nd	⊢ (𝜑 → (2nd ‘(𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹)) = ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ · (2nd ‘𝑍))(2nd ‘𝐹)))

Proof of Theorem xpcco2nd

Step	Hyp	Ref	Expression
1		xpcco1st.t	. . 3 ⊢ 𝑇 = (𝐶 ×c 𝐷)
2		xpcco1st.b	. . 3 ⊢ 𝐵 = (Base‘𝑇)
3		xpcco1st.k	. . 3 ⊢ 𝐾 = (Hom ‘𝑇)
4		eqid 2734	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		xpcco2nd.1	. . 3 ⊢ · = (comp‘𝐷)
6		xpcco1st.o	. . 3 ⊢ 𝑂 = (comp‘𝑇)
7		xpcco1st.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		xpcco1st.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9		xpcco1st.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
10		xpcco1st.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐾𝑌))
11		xpcco1st.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐾𝑍))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	xpcco 18247	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩(comp‘𝐶)(1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ · (2nd ‘𝑍))(2nd ‘𝐹))⟩)
13		ovex 7478	. . 3 ⊢ ((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩(comp‘𝐶)(1st ‘𝑍))(1st ‘𝐹)) ∈ V
14		ovex 7478	. . 3 ⊢ ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ · (2nd ‘𝑍))(2nd ‘𝐹)) ∈ V
15	13, 14	op2ndd 8037	. 2 ⊢ ((𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩(comp‘𝐶)(1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ · (2nd ‘𝑍))(2nd ‘𝐹))⟩ → (2nd ‘(𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹)) = ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ · (2nd ‘𝑍))(2nd ‘𝐹)))
16	12, 15	syl 17	1 ⊢ (𝜑 → (2nd ‘(𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹)) = ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ · (2nd ‘𝑍))(2nd ‘𝐹)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2103  ⟨cop 4654  ‘cfv 6572  (class class class)co 7445  1^st c1st 8024  2^nd c2nd 8025  Basecbs 17253  Hom chom 17317  compcco 17318   ×_c cxpc 18232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-addrcl 11241  ax-mulcl 11242  ax-mulrcl 11243  ax-mulcom 11244  ax-addass 11245  ax-mulass 11246  ax-distr 11247  ax-i2m1 11248  ax-1ne0 11249  ax-1rid 11250  ax-rnegex 11251  ax-rrecex 11252  ax-cnre 11253  ax-pre-lttri 11254  ax-pre-lttrn 11255  ax-pre-ltadd 11256  ax-pre-mulgt0 11257

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-er 8759  df-en 9000  df-dom 9001  df-sdom 9002  df-fin 9003  df-pnf 11322  df-mnf 11323  df-xr 11324  df-ltxr 11325  df-le 11326  df-sub 11518  df-neg 11519  df-nn 12290  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-7 12357  df-8 12358  df-9 12359  df-n0 12550  df-z 12636  df-dec 12755  df-uz 12900  df-fz 13564  df-struct 17189  df-slot 17224  df-ndx 17236  df-base 17254  df-hom 17330  df-cco 17331  df-xpc 18236

This theorem is referenced by:  2ndfcl  18262