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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 6712 | . . 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 17428 | . 2 β’ (π β (πΉ + πΊ) = (π₯ β πΌ β¦ ((πΉβπ₯)(+gβ(π βπ₯))(πΊβπ₯)))) |
11 | 5 | ffvelcdmda 7080 | . . . . 5 β’ ((π β§ π₯ β πΌ) β (π βπ₯) β Mnd) |
12 | 3 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β π β π) |
13 | 4 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β πΌ β π) |
14 | 6 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β π Fn πΌ) |
15 | 7 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β πΉ β π΅) |
16 | simpr 484 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β π₯ β πΌ) | |
17 | 1, 2, 12, 13, 14, 15, 16 | prdsbasprj 17427 | . . . . 5 β’ ((π β§ π₯ β πΌ) β (πΉβπ₯) β (Baseβ(π βπ₯))) |
18 | 8 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β πΌ) β πΊ β π΅) |
19 | 1, 2, 12, 13, 14, 18, 16 | prdsbasprj 17427 | . . . . 5 β’ ((π β§ π₯ β πΌ) β (πΊβπ₯) β (Baseβ(π βπ₯))) |
20 | eqid 2726 | . . . . . 6 β’ (Baseβ(π βπ₯)) = (Baseβ(π βπ₯)) | |
21 | eqid 2726 | . . . . . 6 β’ (+gβ(π βπ₯)) = (+gβ(π βπ₯)) | |
22 | 20, 21 | mndcl 18675 | . . . . 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 1368 | . . . 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 17425 | . . 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 2827 | 1 β’ (π β (πΉ + πΊ) β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 β¦ cmpt 5224 Fn wfn 6532 βΆwf 6533 βcfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 Xscprds 17400 Mndcmnd 18667 |
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-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 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-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-prds 17402 df-mgm 18573 df-sgrp 18652 df-mnd 18668 |
This theorem is referenced by: prdsmndd 18700 prdsrngd 20081 prdsringd 20220 dsmmacl 21636 |
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