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Theorem xpcco1st 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))
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 2737	. . 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 18248	. 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 7471	. . 3 ⊢ ((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)) ∈ V
14		ovex 7471	. . 3 ⊢ ((2^nd ‘𝐺)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝐹)) ∈ V
15	13, 14	op1std 8032	. 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 17	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

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ⟨cop 4640 ‘cfv 6569 (class class class)co 7438 1^st c1st 8020 2^nd c2nd 8021 Basecbs 17254 Hom chom 17318 compcco 17319 ×_c cxpc 18233

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4916 df-iun 5001 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-om 7895 df-1st 8022 df-2nd 8023 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-er 8753 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-nn 12274 df-2 12336 df-3 12337 df-4 12338 df-5 12339 df-6 12340 df-7 12341 df-8 12342 df-9 12343 df-n0 12534 df-z 12621 df-dec 12741 df-uz 12886 df-fz 13554 df-struct 17190 df-slot 17225 df-ndx 17237 df-base 17255 df-hom 17331 df-cco 17332 df-xpc 18237

This theorem is referenced by: 1stfcl 18262

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