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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 2727 | . . 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 18167 | . 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 7447 | . . 3 ⊢ ((1st ‘𝐺)(〈(1st ‘𝑋), (1st ‘𝑌)〉 · (1st ‘𝑍))(1st ‘𝐹)) ∈ V | |
14 | ovex 7447 | . . 3 ⊢ ((2nd ‘𝐺)(〈(2nd ‘𝑋), (2nd ‘𝑌)〉(comp‘𝐷)(2nd ‘𝑍))(2nd ‘𝐹)) ∈ V | |
15 | 13, 14 | op1std 7997 | . 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 1534 ∈ wcel 2099 〈cop 4630 ‘cfv 6542 (class class class)co 7414 1st c1st 7985 2nd c2nd 7986 Basecbs 17173 Hom chom 17237 compcco 17238 ×c cxpc 18152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-slot 17144 df-ndx 17156 df-base 17174 df-hom 17250 df-cco 17251 df-xpc 18156 |
This theorem is referenced by: 1stfcl 18181 |
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