Theorem xpcco2nd 18120

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 2737	. . 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 18118	. 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 7401	. . 3 ⊢ ((1st ‘𝐺)(⟨(1st ‘𝑋), (1st ‘𝑌)⟩(comp‘𝐶)(1st ‘𝑍))(1st ‘𝐹)) ∈ V
14		ovex 7401	. . 3 ⊢ ((2nd ‘𝐺)(⟨(2nd ‘𝑋), (2nd ‘𝑌)⟩ · (2nd ‘𝑍))(2nd ‘𝐹)) ∈ V
15	13, 14	op2ndd 7954	. 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 1542   ∈ wcel 2114  ⟨cop 4588  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  Basecbs 17148  Hom chom 17200  compcco 17201   ×_c cxpc 18103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17149  df-hom 17213  df-cco 17214  df-xpc 18107

This theorem is referenced by:  2ndfcl  18133