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Mirrors > Home > MPE Home > Th. List > prdsplusgcl | Structured version Visualization version GIF version |
Description: Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
Ref | Expression |
---|---|
prdsplusgcl.y | β’ π = (πXsπ ) |
prdsplusgcl.b | β’ π΅ = (Baseβπ) |
prdsplusgcl.p | β’ + = (+gβπ) |
prdsplusgcl.s | β’ (π β π β π) |
prdsplusgcl.i | β’ (π β πΌ β π) |
prdsplusgcl.r | β’ (π β π :πΌβΆMnd) |
prdsplusgcl.f | β’ (π β πΉ β π΅) |
prdsplusgcl.g | β’ (π β πΊ β π΅) |
Ref | Expression |
---|---|
prdsplusgcl | β’ (π β (πΉ + πΊ) β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsplusgcl.y | . . 3 β’ π = (πXsπ ) | |
2 | prdsplusgcl.b | . . 3 β’ π΅ = (Baseβπ) | |
3 | prdsplusgcl.s | . . 3 β’ (π β π β π) | |
4 | prdsplusgcl.i | . . 3 β’ (π β πΌ β π) | |
5 | prdsplusgcl.r | . . . 4 β’ (π β π :πΌβΆMnd) | |
6 | 5 | ffnd 6670 | . . 3 β’ (π β π Fn πΌ) |
7 | prdsplusgcl.f | . . 3 β’ (π β πΉ β π΅) | |
8 | prdsplusgcl.g | . . 3 β’ (π β πΊ β π΅) | |
9 | prdsplusgcl.p | . . 3 β’ + = (+gβπ) | |
10 | 1, 2, 3, 4, 6, 7, 8, 9 | prdsplusgval 17360 | . 2 β’ (π β (πΉ + πΊ) = (π₯ β πΌ β¦ ((πΉβπ₯)(+gβ(π βπ₯))(πΊβπ₯)))) |
11 | 5 | ffvelcdmda 7036 | . . . . 5 β’ ((π β§ π₯ β πΌ) β (π βπ₯) β Mnd) |
12 | 3 | adantr 482 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β π β π) |
13 | 4 | adantr 482 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β πΌ β π) |
14 | 6 | adantr 482 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β π Fn πΌ) |
15 | 7 | adantr 482 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β πΉ β π΅) |
16 | simpr 486 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β π₯ β πΌ) | |
17 | 1, 2, 12, 13, 14, 15, 16 | prdsbasprj 17359 | . . . . 5 β’ ((π β§ π₯ β πΌ) β (πΉβπ₯) β (Baseβ(π βπ₯))) |
18 | 8 | adantr 482 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β πΊ β π΅) |
19 | 1, 2, 12, 13, 14, 18, 16 | prdsbasprj 17359 | . . . . 5 β’ ((π β§ π₯ β πΌ) β (πΊβπ₯) β (Baseβ(π βπ₯))) |
20 | eqid 2733 | . . . . . 6 β’ (Baseβ(π βπ₯)) = (Baseβ(π βπ₯)) | |
21 | eqid 2733 | . . . . . 6 β’ (+gβ(π βπ₯)) = (+gβ(π βπ₯)) | |
22 | 20, 21 | mndcl 18569 | . . . . 5 β’ (((π βπ₯) β Mnd β§ (πΉβπ₯) β (Baseβ(π βπ₯)) β§ (πΊβπ₯) β (Baseβ(π βπ₯))) β ((πΉβπ₯)(+gβ(π βπ₯))(πΊβπ₯)) β (Baseβ(π βπ₯))) |
23 | 11, 17, 19, 22 | syl3anc 1372 | . . . 4 β’ ((π β§ π₯ β πΌ) β ((πΉβπ₯)(+gβ(π βπ₯))(πΊβπ₯)) β (Baseβ(π βπ₯))) |
24 | 23 | ralrimiva 3140 | . . 3 β’ (π β βπ₯ β πΌ ((πΉβπ₯)(+gβ(π βπ₯))(πΊβπ₯)) β (Baseβ(π βπ₯))) |
25 | 1, 2, 3, 4, 6 | prdsbasmpt 17357 | . . 3 β’ (π β ((π₯ β πΌ β¦ ((πΉβπ₯)(+gβ(π βπ₯))(πΊβπ₯))) β π΅ β βπ₯ β πΌ ((πΉβπ₯)(+gβ(π βπ₯))(πΊβπ₯)) β (Baseβ(π βπ₯)))) |
26 | 24, 25 | mpbird 257 | . 2 β’ (π β (π₯ β πΌ β¦ ((πΉβπ₯)(+gβ(π βπ₯))(πΊβπ₯))) β π΅) |
27 | 10, 26 | eqeltrd 2834 | 1 β’ (π β (πΉ + πΊ) β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 β¦ cmpt 5189 Fn wfn 6492 βΆwf 6493 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Xscprds 17332 Mndcmnd 18561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-hom 17162 df-cco 17163 df-prds 17334 df-mgm 18502 df-sgrp 18551 df-mnd 18562 |
This theorem is referenced by: prdsmndd 18594 prdsringd 20041 dsmmacl 21163 |
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