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Theorem xpcco1st 18364

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 18363	. 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 7454	. . 3 ⊢ ((1^st ‘𝐺)(⟨(1^st ‘𝑋), (1^st ‘𝑌)⟩ · (1^st ‘𝑍))(1^st ‘𝐹)) ∈ V
14		ovex 7454	. . 3 ⊢ ((2^nd ‘𝐺)(⟨(2^nd ‘𝑋), (2^nd ‘𝑌)⟩(comp‘𝐷)(2^nd ‘𝑍))(2^nd ‘𝐹)) ∈ V
15	13, 14	op1std 8012	. 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 1569 ∈ wcel 2144 ⟨cop 4834 ‘cfv 6539 (class class class)co 7421 1^st c1st 8000 2^nd c2nd 8001 Basecbs 17393 Hom chom 17445 compcco 17446 ×_c cxpc 18348

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-er 8721 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-fz 13644 df-struct 17331 df-slot 17366 df-ndx 17378 df-base 17394 df-hom 17458 df-cco 17459 df-xpc 18352

This theorem is used by: 1stfcl 18377

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