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Mirrors > Home > MPE Home > Th. List > psrbagaddcl | Structured version Visualization version GIF version |
Description: The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.) Shorten proof and remove a sethood antecedent. (Revised by SN, 7-Aug-2024.) |
Ref | Expression |
---|---|
psrbag.d | β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} |
Ref | Expression |
---|---|
psrbagaddcl | β’ ((πΉ β π· β§ πΊ β π·) β (πΉ βf + πΊ) β π·) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0addcl 12453 | . . . 4 β’ ((π₯ β β0 β§ π¦ β β0) β (π₯ + π¦) β β0) | |
2 | 1 | adantl 483 | . . 3 β’ (((πΉ β π· β§ πΊ β π·) β§ (π₯ β β0 β§ π¦ β β0)) β (π₯ + π¦) β β0) |
3 | psrbag.d | . . . . 5 β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} | |
4 | 3 | psrbagf 21336 | . . . 4 β’ (πΉ β π· β πΉ:πΌβΆβ0) |
5 | 4 | adantr 482 | . . 3 β’ ((πΉ β π· β§ πΊ β π·) β πΉ:πΌβΆβ0) |
6 | 3 | psrbagf 21336 | . . . 4 β’ (πΊ β π· β πΊ:πΌβΆβ0) |
7 | 6 | adantl 483 | . . 3 β’ ((πΉ β π· β§ πΊ β π·) β πΊ:πΌβΆβ0) |
8 | simpl 484 | . . . 4 β’ ((πΉ β π· β§ πΊ β π·) β πΉ β π·) | |
9 | 5 | ffnd 6670 | . . . 4 β’ ((πΉ β π· β§ πΊ β π·) β πΉ Fn πΌ) |
10 | 8, 9 | fndmexd 7844 | . . 3 β’ ((πΉ β π· β§ πΊ β π·) β πΌ β V) |
11 | inidm 4179 | . . 3 β’ (πΌ β© πΌ) = πΌ | |
12 | 2, 5, 7, 10, 10, 11 | off 7636 | . 2 β’ ((πΉ β π· β§ πΊ β π·) β (πΉ βf + πΊ):πΌβΆβ0) |
13 | ovex 7391 | . . . 4 β’ (πΉ βf + πΊ) β V | |
14 | fcdmnn0suppg 12476 | . . . 4 β’ (((πΉ βf + πΊ) β V β§ (πΉ βf + πΊ):πΌβΆβ0) β ((πΉ βf + πΊ) supp 0) = (β‘(πΉ βf + πΊ) β β)) | |
15 | 13, 12, 14 | sylancr 588 | . . 3 β’ ((πΉ β π· β§ πΊ β π·) β ((πΉ βf + πΊ) supp 0) = (β‘(πΉ βf + πΊ) β β)) |
16 | 3 | psrbagfsupp 21338 | . . . . . 6 β’ (πΉ β π· β πΉ finSupp 0) |
17 | 16 | adantr 482 | . . . . 5 β’ ((πΉ β π· β§ πΊ β π·) β πΉ finSupp 0) |
18 | 3 | psrbagfsupp 21338 | . . . . . 6 β’ (πΊ β π· β πΊ finSupp 0) |
19 | 18 | adantl 483 | . . . . 5 β’ ((πΉ β π· β§ πΊ β π·) β πΊ finSupp 0) |
20 | 17, 19 | fsuppunfi 9330 | . . . 4 β’ ((πΉ β π· β§ πΊ β π·) β ((πΉ supp 0) βͺ (πΊ supp 0)) β Fin) |
21 | 0nn0 12433 | . . . . . 6 β’ 0 β β0 | |
22 | 21 | a1i 11 | . . . . 5 β’ ((πΉ β π· β§ πΊ β π·) β 0 β β0) |
23 | 00id 11335 | . . . . . 6 β’ (0 + 0) = 0 | |
24 | 23 | a1i 11 | . . . . 5 β’ ((πΉ β π· β§ πΊ β π·) β (0 + 0) = 0) |
25 | 10, 22, 5, 7, 24 | suppofssd 8135 | . . . 4 β’ ((πΉ β π· β§ πΊ β π·) β ((πΉ βf + πΊ) supp 0) β ((πΉ supp 0) βͺ (πΊ supp 0))) |
26 | 20, 25 | ssfid 9214 | . . 3 β’ ((πΉ β π· β§ πΊ β π·) β ((πΉ βf + πΊ) supp 0) β Fin) |
27 | 15, 26 | eqeltrrd 2835 | . 2 β’ ((πΉ β π· β§ πΊ β π·) β (β‘(πΉ βf + πΊ) β β) β Fin) |
28 | 3 | psrbag 21335 | . . 3 β’ (πΌ β V β ((πΉ βf + πΊ) β π· β ((πΉ βf + πΊ):πΌβΆβ0 β§ (β‘(πΉ βf + πΊ) β β) β Fin))) |
29 | 10, 28 | syl 17 | . 2 β’ ((πΉ β π· β§ πΊ β π·) β ((πΉ βf + πΊ) β π· β ((πΉ βf + πΊ):πΌβΆβ0 β§ (β‘(πΉ βf + πΊ) β β) β Fin))) |
30 | 12, 27, 29 | mpbir2and 712 | 1 β’ ((πΉ β π· β§ πΊ β π·) β (πΉ βf + πΊ) β π·) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3406 Vcvv 3444 βͺ cun 3909 class class class wbr 5106 β‘ccnv 5633 β cima 5637 βΆwf 6493 (class class class)co 7358 βf cof 7616 supp csupp 8093 βm cmap 8768 Fincfn 8886 finSupp cfsupp 9308 0cc0 11056 + caddc 11059 βcn 12158 β0cn0 12418 |
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 |
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-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-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-nn 12159 df-n0 12419 |
This theorem is referenced by: mplmon2mul 21493 evlslem1 21508 tdeglem3 25438 |
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