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Theorem xpcco1st 18272

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))
xpcco1st.1	⊢ · = (comp‘𝐶)

**Assertion**
Ref	Expression
xpcco1st	⊢ (𝜑 → (1^st ‘(𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹)) = ((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)))

**Proof of Theorem xpcco1st**
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		xpcco1st.1	. . 3 ⊢ · = (comp‘𝐶)
5		eqid 2760	. . 3 ⊢ (comp‘𝐷) = (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 18271	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)), ((2^nd ‘𝐺)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝐹))⟩)
13		ovex 7446	. . 3 ⊢ ((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)) ∈ V
14		ovex 7446	. . 3 ⊢ ((2^nd ‘𝐺)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝐹)) ∈ V
15	13, 14	op1std 7996	. 2 ⊢ ((𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹) = ⟨((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)), ((2^nd ‘𝐺)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝐹))⟩ → (1^st ‘(𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹)) = ((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)))
16	12, 15	syl 18	1 ⊢ (𝜑 → (1^st ‘(𝐺(⟨𝑋, 𝑌⟩𝑂𝑍)𝐹)) = ((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ⟨cop 4590 ‘cfv 6533 (class class class)co 7413 1^st c1st 7984 2^nd c2nd 7985 Basecbs 17301 Hom chom 17353 compcco 17354 ×_c cxpc 18256

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-fz 13562 df-struct 17239 df-slot 17274 df-ndx 17286 df-base 17302 df-hom 17366 df-cco 17367 df-xpc 18260

This theorem is used by: 1stfcl 18285

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