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Mirrors > Home > MPE Home > Th. List > xpcco1st | 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)) |
xpcco1st.1 | ⊢ · = (comp‘𝐶) |
Ref | Expression |
---|---|
xpcco1st | ⊢ (𝜑 → (1st ‘(𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹)) = ((1st ‘𝐺)(〈(1st ‘𝑋), (1st ‘𝑌)〉 · (1st ‘𝑍))(1st ‘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 | xpcco1st.1 | . . 3 ⊢ · = (comp‘𝐶) | |
5 | eqid 2772 | . . 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 17281 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = 〈((1st ‘𝐺)(〈(1st ‘𝑋), (1st ‘𝑌)〉 · (1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(〈(2nd ‘𝑋), (2nd ‘𝑌)〉(comp‘𝐷)(2nd ‘𝑍))(2nd ‘𝐹))〉) |
13 | ovex 7002 | . . 3 ⊢ ((1st ‘𝐺)(〈(1st ‘𝑋), (1st ‘𝑌)〉 · (1st ‘𝑍))(1st ‘𝐹)) ∈ V | |
14 | ovex 7002 | . . 3 ⊢ ((2nd ‘𝐺)(〈(2nd ‘𝑋), (2nd ‘𝑌)〉(comp‘𝐷)(2nd ‘𝑍))(2nd ‘𝐹)) ∈ V | |
15 | 13, 14 | op1std 7504 | . 2 ⊢ ((𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = 〈((1st ‘𝐺)(〈(1st ‘𝑋), (1st ‘𝑌)〉 · (1st ‘𝑍))(1st ‘𝐹)), ((2nd ‘𝐺)(〈(2nd ‘𝑋), (2nd ‘𝑌)〉(comp‘𝐷)(2nd ‘𝑍))(2nd ‘𝐹))〉 → (1st ‘(𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹)) = ((1st ‘𝐺)(〈(1st ‘𝑋), (1st ‘𝑌)〉 · (1st ‘𝑍))(1st ‘𝐹))) |
16 | 12, 15 | syl 17 | 1 ⊢ (𝜑 → (1st ‘(𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹)) = ((1st ‘𝐺)(〈(1st ‘𝑋), (1st ‘𝑌)〉 · (1st ‘𝑍))(1st ‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2048 〈cop 4441 ‘cfv 6182 (class class class)co 6970 1st c1st 7492 2nd c2nd 7493 Basecbs 16329 Hom chom 16422 compcco 16423 ×c cxpc 17266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-hom 16435 df-cco 16436 df-xpc 17270 |
This theorem is referenced by: 1stfcl 17295 |
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