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Mirrors > Home > ILE Home > Th. List > hashiun | Unicode version |
Description: The cardinality of a disjoint indexed union. (Contributed by Mario Carneiro, 24-Jan-2015.) (Revised by Mario Carneiro, 10-Dec-2016.) |
Ref | Expression |
---|---|
fsumiun.1 | |
fsumiun.2 | |
fsumiun.3 | Disj |
Ref | Expression |
---|---|
hashiun | ♯ ♯ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumiun.1 | . . 3 | |
2 | fsumiun.2 | . . 3 | |
3 | fsumiun.3 | . . 3 Disj | |
4 | 1cnd 7929 | . . 3 | |
5 | 1, 2, 3, 4 | fsumiun 11433 | . 2 |
6 | 2 | ralrimiva 2543 | . . . . 5 |
7 | iunfidisj 6921 | . . . . 5 Disj | |
8 | 1, 6, 3, 7 | syl3anc 1233 | . . . 4 |
9 | ax-1cn 7860 | . . . 4 | |
10 | fsumconst 11410 | . . . 4 ♯ | |
11 | 8, 9, 10 | sylancl 411 | . . 3 ♯ |
12 | hashcl 10708 | . . . 4 ♯ | |
13 | nn0cn 9138 | . . . 4 ♯ ♯ | |
14 | mulid1 7910 | . . . 4 ♯ ♯ ♯ | |
15 | 8, 12, 13, 14 | 4syl 18 | . . 3 ♯ ♯ |
16 | 11, 15 | eqtrd 2203 | . 2 ♯ |
17 | fsumconst 11410 | . . . . 5 ♯ | |
18 | 2, 9, 17 | sylancl 411 | . . . 4 ♯ |
19 | hashcl 10708 | . . . . 5 ♯ | |
20 | nn0cn 9138 | . . . . 5 ♯ ♯ | |
21 | mulid1 7910 | . . . . 5 ♯ ♯ ♯ | |
22 | 2, 19, 20, 21 | 4syl 18 | . . . 4 ♯ ♯ |
23 | 18, 22 | eqtrd 2203 | . . 3 ♯ |
24 | 23 | sumeq2dv 11324 | . 2 ♯ |
25 | 5, 16, 24 | 3eqtr3d 2211 | 1 ♯ ♯ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wral 2448 ciun 3871 Disj wdisj 3964 cfv 5196 (class class class)co 5851 cfn 6716 cc 7765 c1 7768 cmul 7772 cn0 9128 ♯chash 10702 csu 11309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 ax-arch 7886 ax-caucvg 7887 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-disj 3965 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-frec 6368 df-1o 6393 df-oadd 6397 df-er 6511 df-en 6717 df-dom 6718 df-fin 6719 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-n0 9129 df-z 9206 df-uz 9481 df-q 9572 df-rp 9604 df-fz 9959 df-fzo 10092 df-seqfrec 10395 df-exp 10469 df-ihash 10703 df-cj 10799 df-re 10800 df-im 10801 df-rsqrt 10955 df-abs 10956 df-clim 11235 df-sumdc 11310 |
This theorem is referenced by: hash2iun 11435 hashrabrex 11437 hashuni 11438 phisum 12187 |
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