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Mirrors > Home > MPE Home > Th. List > xpcco2nd | Structured version Visualization version GIF version |
Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
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) |
Ref | Expression |
---|---|
xpcco2nd | ⊢ (𝜑 → (2nd ‘(𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹)) = ((2nd ‘𝐺)(〈(2nd ‘𝑋), (2nd ‘𝑌)〉 · (2nd ‘𝑍))(2nd ‘𝐹))) |
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 2610 | . . 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 16646 | . 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 6577 | . . 3 ⊢ ((1st ‘𝐺)(〈(1st ‘𝑋), (1st ‘𝑌)〉(comp‘𝐶)(1st ‘𝑍))(1st ‘𝐹)) ∈ V | |
14 | ovex 6577 | . . 3 ⊢ ((2nd ‘𝐺)(〈(2nd ‘𝑋), (2nd ‘𝑌)〉 · (2nd ‘𝑍))(2nd ‘𝐹)) ∈ V | |
15 | 13, 14 | op2ndd 7070 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 Basecbs 15695 Hom chom 15779 compcco 15780 ×c cxpc 16631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-hom 15793 df-cco 15794 df-xpc 16635 |
This theorem is referenced by: 2ndfcl 16661 |
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