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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 17428 | . . 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 5593 | . . 3 ⊢ (𝜑 → 〈𝑀, 𝑁〉 ∈ (𝑋 × 𝑌)) |
11 | xpcco2.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
12 | xpcco2.q | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑌) | |
13 | 11, 12 | opelxpd 5593 | . . 3 ⊢ (𝜑 → 〈𝑃, 𝑄〉 ∈ (𝑋 × 𝑌)) |
14 | 1, 4, 5, 6, 7, 10, 13 | xpchom 17430 | . 2 ⊢ (𝜑 → (〈𝑀, 𝑁〉𝐾〈𝑃, 𝑄〉) = (((1st ‘〈𝑀, 𝑁〉)𝐻(1st ‘〈𝑃, 𝑄〉)) × ((2nd ‘〈𝑀, 𝑁〉)𝐽(2nd ‘〈𝑃, 𝑄〉)))) |
15 | op1stg 7701 | . . . . 5 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (1st ‘〈𝑀, 𝑁〉) = 𝑀) | |
16 | 8, 9, 15 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (1st ‘〈𝑀, 𝑁〉) = 𝑀) |
17 | op1stg 7701 | . . . . 5 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (1st ‘〈𝑃, 𝑄〉) = 𝑃) | |
18 | 11, 12, 17 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (1st ‘〈𝑃, 𝑄〉) = 𝑃) |
19 | 16, 18 | oveq12d 7174 | . . 3 ⊢ (𝜑 → ((1st ‘〈𝑀, 𝑁〉)𝐻(1st ‘〈𝑃, 𝑄〉)) = (𝑀𝐻𝑃)) |
20 | op2ndg 7702 | . . . . 5 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (2nd ‘〈𝑀, 𝑁〉) = 𝑁) | |
21 | 8, 9, 20 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (2nd ‘〈𝑀, 𝑁〉) = 𝑁) |
22 | op2ndg 7702 | . . . . 5 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌) → (2nd ‘〈𝑃, 𝑄〉) = 𝑄) | |
23 | 11, 12, 22 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (2nd ‘〈𝑃, 𝑄〉) = 𝑄) |
24 | 21, 23 | oveq12d 7174 | . . 3 ⊢ (𝜑 → ((2nd ‘〈𝑀, 𝑁〉)𝐽(2nd ‘〈𝑃, 𝑄〉)) = (𝑁𝐽𝑄)) |
25 | 19, 24 | xpeq12d 5586 | . 2 ⊢ (𝜑 → (((1st ‘〈𝑀, 𝑁〉)𝐻(1st ‘〈𝑃, 𝑄〉)) × ((2nd ‘〈𝑀, 𝑁〉)𝐽(2nd ‘〈𝑃, 𝑄〉))) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄))) |
26 | 14, 25 | eqtrd 2856 | 1 ⊢ (𝜑 → (〈𝑀, 𝑁〉𝐾〈𝑃, 𝑄〉) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cop 4573 × cxp 5553 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 2nd c2nd 7688 Basecbs 16483 Hom chom 16576 ×c cxpc 17418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-hom 16589 df-cco 16590 df-xpc 17422 |
This theorem is referenced by: xpcco2 17437 prfcl 17453 evlfcl 17472 curf1cl 17478 curf2cl 17481 curfcl 17482 uncf2 17487 uncfcurf 17489 diag12 17494 diag2 17495 curf2ndf 17497 yonedalem22 17528 yonedalem3b 17529 |
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