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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 2733 | . . 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 17928 | . 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 7328 | . . 3 ⊢ ((1st ‘𝐺)(〈(1st ‘𝑋), (1st ‘𝑌)〉(comp‘𝐶)(1st ‘𝑍))(1st ‘𝐹)) ∈ V | |
14 | ovex 7328 | . . 3 ⊢ ((2nd ‘𝐺)(〈(2nd ‘𝑋), (2nd ‘𝑌)〉 · (2nd ‘𝑍))(2nd ‘𝐹)) ∈ V | |
15 | 13, 14 | op2ndd 7862 | . 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 1537 ∈ wcel 2101 〈cop 4570 ‘cfv 6447 (class class class)co 7295 1st c1st 7849 2nd c2nd 7850 Basecbs 16940 Hom chom 17001 compcco 17002 ×c cxpc 17913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-fz 13268 df-struct 16876 df-slot 16911 df-ndx 16923 df-base 16941 df-hom 17014 df-cco 17015 df-xpc 17917 |
This theorem is referenced by: 2ndfcl 17943 |
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