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Mirrors > Home > MPE Home > Th. List > xpchom2 | Structured version Visualization version GIF version |
Description: Value of the set of morphisms in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
xpcco2.t | ⊢ 𝑇 = (𝐶 ×c 𝐷) |
xpcco2.x | ⊢ 𝑋 = (Base‘𝐶) |
xpcco2.y | ⊢ 𝑌 = (Base‘𝐷) |
xpcco2.h | ⊢ 𝐻 = (Hom ‘𝐶) |
xpcco2.j | ⊢ 𝐽 = (Hom ‘𝐷) |
xpcco2.m | ⊢ (𝜑 → 𝑀 ∈ 𝑋) |
xpcco2.n | ⊢ (𝜑 → 𝑁 ∈ 𝑌) |
xpcco2.p | ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
xpcco2.q | ⊢ (𝜑 → 𝑄 ∈ 𝑌) |
xpchom2.k | ⊢ 𝐾 = (Hom ‘𝑇) |
Ref | Expression |
---|---|
xpchom2 | ⊢ (𝜑 → (〈𝑀, 𝑁〉𝐾〈𝑃, 𝑄〉) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpcco2.t | . . 3 ⊢ 𝑇 = (𝐶 ×c 𝐷) | |
2 | xpcco2.x | . . . 4 ⊢ 𝑋 = (Base‘𝐶) | |
3 | xpcco2.y | . . . 4 ⊢ 𝑌 = (Base‘𝐷) | |
4 | 1, 2, 3 | xpcbas 16641 | . . 3 ⊢ (𝑋 × 𝑌) = (Base‘𝑇) |
5 | xpcco2.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
6 | xpcco2.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
7 | xpchom2.k | . . 3 ⊢ 𝐾 = (Hom ‘𝑇) | |
8 | xpcco2.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑋) | |
9 | xpcco2.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑌) | |
10 | opelxpi 5072 | . . . 4 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → 〈𝑀, 𝑁〉 ∈ (𝑋 × 𝑌)) | |
11 | 8, 9, 10 | syl2anc 691 | . . 3 ⊢ (𝜑 → 〈𝑀, 𝑁〉 ∈ (𝑋 × 𝑌)) |
12 | xpcco2.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
13 | xpcco2.q | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑌) | |
14 | opelxpi 5072 | . . . 4 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → 〈𝑃, 𝑄〉 ∈ (𝑋 × 𝑌)) | |
15 | 12, 13, 14 | syl2anc 691 | . . 3 ⊢ (𝜑 → 〈𝑃, 𝑄〉 ∈ (𝑋 × 𝑌)) |
16 | 1, 4, 5, 6, 7, 11, 15 | xpchom 16643 | . 2 ⊢ (𝜑 → (〈𝑀, 𝑁〉𝐾〈𝑃, 𝑄〉) = (((1st ‘〈𝑀, 𝑁〉)𝐻(1st ‘〈𝑃, 𝑄〉)) × ((2nd ‘〈𝑀, 𝑁〉)𝐽(2nd ‘〈𝑃, 𝑄〉)))) |
17 | op1stg 7071 | . . . . 5 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (1st ‘〈𝑀, 𝑁〉) = 𝑀) | |
18 | 8, 9, 17 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (1st ‘〈𝑀, 𝑁〉) = 𝑀) |
19 | op1stg 7071 | . . . . 5 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (1st ‘〈𝑃, 𝑄〉) = 𝑃) | |
20 | 12, 13, 19 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (1st ‘〈𝑃, 𝑄〉) = 𝑃) |
21 | 18, 20 | oveq12d 6567 | . . 3 ⊢ (𝜑 → ((1st ‘〈𝑀, 𝑁〉)𝐻(1st ‘〈𝑃, 𝑄〉)) = (𝑀𝐻𝑃)) |
22 | op2ndg 7072 | . . . . 5 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (2nd ‘〈𝑀, 𝑁〉) = 𝑁) | |
23 | 8, 9, 22 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (2nd ‘〈𝑀, 𝑁〉) = 𝑁) |
24 | op2ndg 7072 | . . . . 5 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (2nd ‘〈𝑃, 𝑄〉) = 𝑄) | |
25 | 12, 13, 24 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (2nd ‘〈𝑃, 𝑄〉) = 𝑄) |
26 | 23, 25 | oveq12d 6567 | . . 3 ⊢ (𝜑 → ((2nd ‘〈𝑀, 𝑁〉)𝐽(2nd ‘〈𝑃, 𝑄〉)) = (𝑁𝐽𝑄)) |
27 | 21, 26 | xpeq12d 5064 | . 2 ⊢ (𝜑 → (((1st ‘〈𝑀, 𝑁〉)𝐻(1st ‘〈𝑃, 𝑄〉)) × ((2nd ‘〈𝑀, 𝑁〉)𝐽(2nd ‘〈𝑃, 𝑄〉))) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄))) |
28 | 16, 27 | eqtrd 2644 | 1 ⊢ (𝜑 → (〈𝑀, 𝑁〉𝐾〈𝑃, 𝑄〉) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 Basecbs 15695 Hom chom 15779 ×c cxpc 16631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-hom 15793 df-cco 15794 df-xpc 16635 |
This theorem is referenced by: xpcco2 16650 prfcl 16666 evlfcl 16685 curf1cl 16691 curf2cl 16694 curfcl 16695 uncf2 16700 uncfcurf 16702 diag12 16707 diag2 16708 curf2ndf 16710 yonedalem22 16741 yonedalem3b 16742 |
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