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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 486 | . . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑢 = 𝑋) | |
4 | 3 | fveq2d 6680 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1st ‘𝑢) = (1st ‘𝑋)) |
5 | simpr 488 | . . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑣 = 𝑌) | |
6 | 5 | fveq2d 6680 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1st ‘𝑣) = (1st ‘𝑌)) |
7 | 4, 6 | oveq12d 7190 | . . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((1st ‘𝑢)𝐻(1st ‘𝑣)) = ((1st ‘𝑋)𝐻(1st ‘𝑌))) |
8 | 3 | fveq2d 6680 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2nd ‘𝑢) = (2nd ‘𝑋)) |
9 | 5 | fveq2d 6680 | . . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2nd ‘𝑣) = (2nd ‘𝑌)) |
10 | 8, 9 | oveq12d 7190 | . . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((2nd ‘𝑢)𝐽(2nd ‘𝑣)) = ((2nd ‘𝑋)𝐽(2nd ‘𝑌))) |
11 | 7, 10 | xpeq12d 5556 | . . 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 17547 | . . 3 ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣)))) |
18 | ovex 7205 | . . . 4 ⊢ ((1st ‘𝑋)𝐻(1st ‘𝑌)) ∈ V | |
19 | ovex 7205 | . . . 4 ⊢ ((2nd ‘𝑋)𝐽(2nd ‘𝑌)) ∈ V | |
20 | 18, 19 | xpex 7496 | . . 3 ⊢ (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌))) ∈ V |
21 | 11, 17, 20 | ovmpoa 7322 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌)))) |
22 | 1, 2, 21 | syl2anc 587 | 1 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 × cxp 5523 ‘cfv 6339 (class class class)co 7172 1st c1st 7714 2nd c2nd 7715 Basecbs 16588 Hom chom 16681 ×c cxpc 17536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-z 12065 df-dec 12182 df-uz 12327 df-fz 12984 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-hom 16694 df-cco 16695 df-xpc 17540 |
This theorem is referenced by: xpchom2 17554 xpccatid 17556 1stfcl 17565 2ndfcl 17566 xpcpropd 17576 evlfcl 17590 curf1cl 17596 hofcl 17627 yonedalem3 17648 |
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