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Mirrors > Home > MPE Home > Th. List > xpcco1st | Structured version Visualization version GIF version |
Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
xpcco1st.t | โข ๐ = (๐ถ รc ๐ท) |
xpcco1st.b | โข ๐ต = (Baseโ๐) |
xpcco1st.k | โข ๐พ = (Hom โ๐) |
xpcco1st.o | โข ๐ = (compโ๐) |
xpcco1st.x | โข (๐ โ ๐ โ ๐ต) |
xpcco1st.y | โข (๐ โ ๐ โ ๐ต) |
xpcco1st.z | โข (๐ โ ๐ โ ๐ต) |
xpcco1st.f | โข (๐ โ ๐น โ (๐๐พ๐)) |
xpcco1st.g | โข (๐ โ ๐บ โ (๐๐พ๐)) |
xpcco1st.1 | โข ยท = (compโ๐ถ) |
Ref | Expression |
---|---|
xpcco1st | โข (๐ โ (1st โ(๐บ(โจ๐, ๐โฉ๐๐)๐น)) = ((1st โ๐บ)(โจ(1st โ๐), (1st โ๐)โฉ ยท (1st โ๐))(1st โ๐น))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpcco1st.t | . . 3 โข ๐ = (๐ถ รc ๐ท) | |
2 | xpcco1st.b | . . 3 โข ๐ต = (Baseโ๐) | |
3 | xpcco1st.k | . . 3 โข ๐พ = (Hom โ๐) | |
4 | xpcco1st.1 | . . 3 โข ยท = (compโ๐ถ) | |
5 | eqid 2726 | . . 3 โข (compโ๐ท) = (compโ๐ท) | |
6 | xpcco1st.o | . . 3 โข ๐ = (compโ๐) | |
7 | xpcco1st.x | . . 3 โข (๐ โ ๐ โ ๐ต) | |
8 | xpcco1st.y | . . 3 โข (๐ โ ๐ โ ๐ต) | |
9 | xpcco1st.z | . . 3 โข (๐ โ ๐ โ ๐ต) | |
10 | xpcco1st.f | . . 3 โข (๐ โ ๐น โ (๐๐พ๐)) | |
11 | xpcco1st.g | . . 3 โข (๐ โ ๐บ โ (๐๐พ๐)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | xpcco 18147 | . 2 โข (๐ โ (๐บ(โจ๐, ๐โฉ๐๐)๐น) = โจ((1st โ๐บ)(โจ(1st โ๐), (1st โ๐)โฉ ยท (1st โ๐))(1st โ๐น)), ((2nd โ๐บ)(โจ(2nd โ๐), (2nd โ๐)โฉ(compโ๐ท)(2nd โ๐))(2nd โ๐น))โฉ) |
13 | ovex 7438 | . . 3 โข ((1st โ๐บ)(โจ(1st โ๐), (1st โ๐)โฉ ยท (1st โ๐))(1st โ๐น)) โ V | |
14 | ovex 7438 | . . 3 โข ((2nd โ๐บ)(โจ(2nd โ๐), (2nd โ๐)โฉ(compโ๐ท)(2nd โ๐))(2nd โ๐น)) โ V | |
15 | 13, 14 | op1std 7984 | . 2 โข ((๐บ(โจ๐, ๐โฉ๐๐)๐น) = โจ((1st โ๐บ)(โจ(1st โ๐), (1st โ๐)โฉ ยท (1st โ๐))(1st โ๐น)), ((2nd โ๐บ)(โจ(2nd โ๐), (2nd โ๐)โฉ(compโ๐ท)(2nd โ๐))(2nd โ๐น))โฉ โ (1st โ(๐บ(โจ๐, ๐โฉ๐๐)๐น)) = ((1st โ๐บ)(โจ(1st โ๐), (1st โ๐)โฉ ยท (1st โ๐))(1st โ๐น))) |
16 | 12, 15 | syl 17 | 1 โข (๐ โ (1st โ(๐บ(โจ๐, ๐โฉ๐๐)๐น)) = ((1st โ๐บ)(โจ(1st โ๐), (1st โ๐)โฉ ยท (1st โ๐))(1st โ๐น))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โจcop 4629 โcfv 6537 (class class class)co 7405 1st c1st 7972 2nd c2nd 7973 Basecbs 17153 Hom chom 17217 compcco 17218 รc cxpc 18132 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-hom 17230 df-cco 17231 df-xpc 18136 |
This theorem is referenced by: 1stfcl 18161 |
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