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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 482 | . . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑢 = 𝑋) | |
4 | 3 | fveq2d 6888 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1st ‘𝑢) = (1st ‘𝑋)) |
5 | simpr 484 | . . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑣 = 𝑌) | |
6 | 5 | fveq2d 6888 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1st ‘𝑣) = (1st ‘𝑌)) |
7 | 4, 6 | oveq12d 7422 | . . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((1st ‘𝑢)𝐻(1st ‘𝑣)) = ((1st ‘𝑋)𝐻(1st ‘𝑌))) |
8 | 3 | fveq2d 6888 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2nd ‘𝑢) = (2nd ‘𝑋)) |
9 | 5 | fveq2d 6888 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2nd ‘𝑣) = (2nd ‘𝑌)) |
10 | 8, 9 | oveq12d 7422 | . . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((2nd ‘𝑢)𝐽(2nd ‘𝑣)) = ((2nd ‘𝑋)𝐽(2nd ‘𝑌))) |
11 | 7, 10 | xpeq12d 5700 | . . 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 18141 | . . 3 ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣)))) |
18 | ovex 7437 | . . . 4 ⊢ ((1st ‘𝑋)𝐻(1st ‘𝑌)) ∈ V | |
19 | ovex 7437 | . . . 4 ⊢ ((2nd ‘𝑋)𝐽(2nd ‘𝑌)) ∈ V | |
20 | 18, 19 | xpex 7736 | . . 3 ⊢ (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌))) ∈ V |
21 | 11, 17, 20 | ovmpoa 7558 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌)))) |
22 | 1, 2, 21 | syl2anc 583 | 1 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 × cxp 5667 ‘cfv 6536 (class class class)co 7404 1st c1st 7969 2nd c2nd 7970 Basecbs 17151 Hom chom 17215 ×c cxpc 18130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-hom 17228 df-cco 17229 df-xpc 18134 |
This theorem is referenced by: xpchom2 18148 xpccatid 18150 1stfcl 18159 2ndfcl 18160 xpcpropd 18171 evlfcl 18185 curf1cl 18191 hofcl 18222 yonedalem3 18243 |
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