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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 17431 | . . 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 5596 | . . 3 ⊢ (𝜑 → 〈𝑀, 𝑁〉 ∈ (𝑋 × 𝑌)) |
11 | xpcco2.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
12 | xpcco2.q | . . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝑌) | |
13 | 11, 12 | opelxpd 5596 | . . 3 ⊢ (𝜑 → 〈𝑃, 𝑄〉 ∈ (𝑋 × 𝑌)) |
14 | 1, 4, 5, 6, 7, 10, 13 | xpchom 17433 | . 2 ⊢ (𝜑 → (〈𝑀, 𝑁〉𝐾〈𝑃, 𝑄〉) = (((1st ‘〈𝑀, 𝑁〉)𝐻(1st ‘〈𝑃, 𝑄〉)) × ((2nd ‘〈𝑀, 𝑁〉)𝐽(2nd ‘〈𝑃, 𝑄〉)))) |
15 | op1stg 7704 | . . . . 5 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (1st ‘〈𝑀, 𝑁〉) = 𝑀) | |
16 | 8, 9, 15 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (1st ‘〈𝑀, 𝑁〉) = 𝑀) |
17 | op1stg 7704 | . . . . 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 7177 | . . 3 ⊢ (𝜑 → ((1st ‘〈𝑀, 𝑁〉)𝐻(1st ‘〈𝑃, 𝑄〉)) = (𝑀𝐻𝑃)) |
20 | op2ndg 7705 | . . . . 5 ⊢ ((𝑀 ∈ 𝑋 ∧ 𝑁 ∈ 𝑌) → (2nd ‘〈𝑀, 𝑁〉) = 𝑁) | |
21 | 8, 9, 20 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (2nd ‘〈𝑀, 𝑁〉) = 𝑁) |
22 | op2ndg 7705 | . . . . 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 7177 | . . 3 ⊢ (𝜑 → ((2nd ‘〈𝑀, 𝑁〉)𝐽(2nd ‘〈𝑃, 𝑄〉)) = (𝑁𝐽𝑄)) |
25 | 19, 24 | xpeq12d 5589 | . 2 ⊢ (𝜑 → (((1st ‘〈𝑀, 𝑁〉)𝐻(1st ‘〈𝑃, 𝑄〉)) × ((2nd ‘〈𝑀, 𝑁〉)𝐽(2nd ‘〈𝑃, 𝑄〉))) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄))) |
26 | 14, 25 | eqtrd 2859 | 1 ⊢ (𝜑 → (〈𝑀, 𝑁〉𝐾〈𝑃, 𝑄〉) = ((𝑀𝐻𝑃) × (𝑁𝐽𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 〈cop 4576 × cxp 5556 ‘cfv 6358 (class class class)co 7159 1st c1st 7690 2nd c2nd 7691 Basecbs 16486 Hom chom 16579 ×c cxpc 17421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-hom 16592 df-cco 16593 df-xpc 17425 |
This theorem is referenced by: xpcco2 17440 prfcl 17456 evlfcl 17475 curf1cl 17481 curf2cl 17484 curfcl 17485 uncf2 17490 uncfcurf 17492 diag12 17497 diag2 17498 curf2ndf 17500 yonedalem22 17531 yonedalem3b 17532 |
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