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Mirrors > Home > MPE Home > Th. List > prdsmulrcl | Structured version Visualization version GIF version |
Description: A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.) |
Ref | Expression |
---|---|
prdsmulrcl.y | β’ π = (πXsπ ) |
prdsmulrcl.b | β’ π΅ = (Baseβπ) |
prdsmulrcl.t | β’ Β· = (.rβπ) |
prdsmulrcl.s | β’ (π β π β π) |
prdsmulrcl.i | β’ (π β πΌ β π) |
prdsmulrcl.r | β’ (π β π :πΌβΆRing) |
prdsmulrcl.f | β’ (π β πΉ β π΅) |
prdsmulrcl.g | β’ (π β πΊ β π΅) |
Ref | Expression |
---|---|
prdsmulrcl | β’ (π β (πΉ Β· πΊ) β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsmulrcl.y | . . 3 β’ π = (πXsπ ) | |
2 | prdsmulrcl.b | . . 3 β’ π΅ = (Baseβπ) | |
3 | prdsmulrcl.s | . . 3 β’ (π β π β π) | |
4 | prdsmulrcl.i | . . 3 β’ (π β πΌ β π) | |
5 | prdsmulrcl.r | . . . 4 β’ (π β π :πΌβΆRing) | |
6 | 5 | ffnd 6718 | . . 3 β’ (π β π Fn πΌ) |
7 | prdsmulrcl.f | . . 3 β’ (π β πΉ β π΅) | |
8 | prdsmulrcl.g | . . 3 β’ (π β πΊ β π΅) | |
9 | prdsmulrcl.t | . . 3 β’ Β· = (.rβπ) | |
10 | 1, 2, 3, 4, 6, 7, 8, 9 | prdsmulrval 17420 | . 2 β’ (π β (πΉ Β· πΊ) = (π₯ β πΌ β¦ ((πΉβπ₯)(.rβ(π βπ₯))(πΊβπ₯)))) |
11 | 5 | ffvelcdmda 7086 | . . . . 5 β’ ((π β§ π₯ β πΌ) β (π βπ₯) β Ring) |
12 | 3 | adantr 481 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β π β π) |
13 | 4 | adantr 481 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β πΌ β π) |
14 | 6 | adantr 481 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β π Fn πΌ) |
15 | 7 | adantr 481 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β πΉ β π΅) |
16 | simpr 485 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β π₯ β πΌ) | |
17 | 1, 2, 12, 13, 14, 15, 16 | prdsbasprj 17417 | . . . . 5 β’ ((π β§ π₯ β πΌ) β (πΉβπ₯) β (Baseβ(π βπ₯))) |
18 | 8 | adantr 481 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β πΊ β π΅) |
19 | 1, 2, 12, 13, 14, 18, 16 | prdsbasprj 17417 | . . . . 5 β’ ((π β§ π₯ β πΌ) β (πΊβπ₯) β (Baseβ(π βπ₯))) |
20 | eqid 2732 | . . . . . 6 β’ (Baseβ(π βπ₯)) = (Baseβ(π βπ₯)) | |
21 | eqid 2732 | . . . . . 6 β’ (.rβ(π βπ₯)) = (.rβ(π βπ₯)) | |
22 | 20, 21 | ringcl 20072 | . . . . 5 β’ (((π βπ₯) β Ring β§ (πΉβπ₯) β (Baseβ(π βπ₯)) β§ (πΊβπ₯) β (Baseβ(π βπ₯))) β ((πΉβπ₯)(.rβ(π βπ₯))(πΊβπ₯)) β (Baseβ(π βπ₯))) |
23 | 11, 17, 19, 22 | syl3anc 1371 | . . . 4 β’ ((π β§ π₯ β πΌ) β ((πΉβπ₯)(.rβ(π βπ₯))(πΊβπ₯)) β (Baseβ(π βπ₯))) |
24 | 23 | ralrimiva 3146 | . . 3 β’ (π β βπ₯ β πΌ ((πΉβπ₯)(.rβ(π βπ₯))(πΊβπ₯)) β (Baseβ(π βπ₯))) |
25 | 1, 2, 3, 4, 6 | prdsbasmpt 17415 | . . 3 β’ (π β ((π₯ β πΌ β¦ ((πΉβπ₯)(.rβ(π βπ₯))(πΊβπ₯))) β π΅ β βπ₯ β πΌ ((πΉβπ₯)(.rβ(π βπ₯))(πΊβπ₯)) β (Baseβ(π βπ₯)))) |
26 | 24, 25 | mpbird 256 | . 2 β’ (π β (π₯ β πΌ β¦ ((πΉβπ₯)(.rβ(π βπ₯))(πΊβπ₯))) β π΅) |
27 | 10, 26 | eqeltrd 2833 | 1 β’ (π β (πΉ Β· πΊ) β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 β¦ cmpt 5231 Fn wfn 6538 βΆwf 6539 βcfv 6543 (class class class)co 7408 Basecbs 17143 .rcmulr 17197 Xscprds 17390 Ringcrg 20055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-prds 17392 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mgp 19987 df-ring 20057 |
This theorem is referenced by: prdsringd 20133 |
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