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Mirrors > Home > MPE Home > Th. List > xpchom | Structured version Visualization version GIF version |
Description: Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
xpchomfval.t | ⊢ 𝑇 = (𝐶 ×c 𝐷) |
xpchomfval.y | ⊢ 𝐵 = (Base‘𝑇) |
xpchomfval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
xpchomfval.j | ⊢ 𝐽 = (Hom ‘𝐷) |
xpchomfval.k | ⊢ 𝐾 = (Hom ‘𝑇) |
xpchom.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
xpchom.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
xpchom | ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpchom.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | xpchom.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
3 | simpl 483 | . . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑢 = 𝑋) | |
4 | 3 | fveq2d 6778 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1st ‘𝑢) = (1st ‘𝑋)) |
5 | simpr 485 | . . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑣 = 𝑌) | |
6 | 5 | fveq2d 6778 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1st ‘𝑣) = (1st ‘𝑌)) |
7 | 4, 6 | oveq12d 7293 | . . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((1st ‘𝑢)𝐻(1st ‘𝑣)) = ((1st ‘𝑋)𝐻(1st ‘𝑌))) |
8 | 3 | fveq2d 6778 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2nd ‘𝑢) = (2nd ‘𝑋)) |
9 | 5 | fveq2d 6778 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2nd ‘𝑣) = (2nd ‘𝑌)) |
10 | 8, 9 | oveq12d 7293 | . . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((2nd ‘𝑢)𝐽(2nd ‘𝑣)) = ((2nd ‘𝑋)𝐽(2nd ‘𝑌))) |
11 | 7, 10 | xpeq12d 5620 | . . 3 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣))) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌)))) |
12 | xpchomfval.t | . . . 4 ⊢ 𝑇 = (𝐶 ×c 𝐷) | |
13 | xpchomfval.y | . . . 4 ⊢ 𝐵 = (Base‘𝑇) | |
14 | xpchomfval.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
15 | xpchomfval.j | . . . 4 ⊢ 𝐽 = (Hom ‘𝐷) | |
16 | xpchomfval.k | . . . 4 ⊢ 𝐾 = (Hom ‘𝑇) | |
17 | 12, 13, 14, 15, 16 | xpchomfval 17896 | . . 3 ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣)))) |
18 | ovex 7308 | . . . 4 ⊢ ((1st ‘𝑋)𝐻(1st ‘𝑌)) ∈ V | |
19 | ovex 7308 | . . . 4 ⊢ ((2nd ‘𝑋)𝐽(2nd ‘𝑌)) ∈ V | |
20 | 18, 19 | xpex 7603 | . . 3 ⊢ (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌))) ∈ V |
21 | 11, 17, 20 | ovmpoa 7428 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌)))) |
22 | 1, 2, 21 | syl2anc 584 | 1 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 × cxp 5587 ‘cfv 6433 (class class class)co 7275 1st c1st 7829 2nd c2nd 7830 Basecbs 16912 Hom chom 16973 ×c cxpc 17885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-hom 16986 df-cco 16987 df-xpc 17889 |
This theorem is referenced by: xpchom2 17903 xpccatid 17905 1stfcl 17914 2ndfcl 17915 xpcpropd 17926 evlfcl 17940 curf1cl 17946 hofcl 17977 yonedalem3 17998 |
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