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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 18174 | . . 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 | 8, 9 | opelxpd 5719 | . . 3 ⊢ (𝜑 → ⟨𝑀, 𝑁⟩ ∈ (𝑋 × 𝑌)) |
11 | xpcco2.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
12 | xpcco2.q | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑌) | |
13 | 11, 12 | opelxpd 5719 | . . 3 ⊢ (𝜑 → ⟨𝑃, 𝑄⟩ ∈ (𝑋 × 𝑌)) |
14 | 1, 4, 5, 6, 7, 10, 13 | xpchom 18176 | . 2 ⊢ (𝜑 → (⟨𝑀, 𝑁⟩𝐾⟨𝑃, 𝑄⟩) = (((1st ‘⟨𝑀, 𝑁⟩)𝐻(1st ‘⟨𝑃, 𝑄⟩)) × ((2nd ‘⟨𝑀, 𝑁⟩)𝐽(2nd ‘⟨𝑃, 𝑄⟩)))) |
15 | op1stg 8009 | . . . . 5 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀) | |
16 | 8, 9, 15 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀) |
17 | op1stg 8009 | . . . . 5 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃) | |
18 | 11, 12, 17 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (1st ‘⟨𝑃, 𝑄⟩) = 𝑃) |
19 | 16, 18 | oveq12d 7442 | . . 3 ⊢ (𝜑 → ((1st ‘⟨𝑀, 𝑁⟩)𝐻(1st ‘⟨𝑃, 𝑄⟩)) = (𝑀𝐻𝑃)) |
20 | op2ndg 8010 | . . . . 5 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁) | |
21 | 8, 9, 20 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁) |
22 | op2ndg 8010 | . . . . 5 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄) | |
23 | 11, 12, 22 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (2nd ‘⟨𝑃, 𝑄⟩) = 𝑄) |
24 | 21, 23 | oveq12d 7442 | . . 3 ⊢ (𝜑 → ((2nd ‘⟨𝑀, 𝑁⟩)𝐽(2nd ‘⟨𝑃, 𝑄⟩)) = (𝑁𝐽𝑄)) |
25 | 19, 24 | xpeq12d 5711 | . 2 ⊢ (𝜑 → (((1st ‘⟨𝑀, 𝑁⟩)𝐻(1st ‘⟨𝑃, 𝑄⟩)) × ((2nd ‘⟨𝑀, 𝑁⟩)𝐽(2nd ‘⟨𝑃, 𝑄⟩))) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄))) |
26 | 14, 25 | eqtrd 2767 | 1 ⊢ (𝜑 → (⟨𝑀, 𝑁⟩𝐾⟨𝑃, 𝑄⟩) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⟨cop 4636 × cxp 5678 ‘cfv 6551 (class class class)co 7424 1st c1st 7995 2nd c2nd 7996 Basecbs 17185 Hom chom 17249 ×c cxpc 18164 |
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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-fz 13523 df-struct 17121 df-slot 17156 df-ndx 17168 df-base 17186 df-hom 17262 df-cco 17263 df-xpc 18168 |
This theorem is referenced by: xpcco2 18183 prfcl 18199 evlfcl 18219 curf1cl 18225 curf2cl 18228 curfcl 18229 uncf2 18234 uncfcurf 18236 diag12 18241 diag2 18242 curf2ndf 18244 yonedalem22 18275 yonedalem3b 18276 |
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