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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovnsubadd2 | Structured version Visualization version GIF version |
Description: (voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . The special case of the union of 2 sets. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
ovnsubadd2.x | ⊢ (𝜑 → 𝑋 ∈ Fin) |
ovnsubadd2.a | ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) |
ovnsubadd2.b | ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑m 𝑋)) |
Ref | Expression |
---|---|
ovnsubadd2 | ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovnsubadd2.x | . 2 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
2 | ovnsubadd2.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) | |
3 | ovnsubadd2.b | . 2 ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑m 𝑋)) | |
4 | eqeq1 2822 | . . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 = 1 ↔ 𝑛 = 1)) | |
5 | eqeq1 2822 | . . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚 = 2 ↔ 𝑛 = 2)) | |
6 | 5 | ifbid 4485 | . . . 4 ⊢ (𝑚 = 𝑛 → if(𝑚 = 2, 𝐵, ∅) = if(𝑛 = 2, 𝐵, ∅)) |
7 | 4, 6 | ifbieq2d 4488 | . . 3 ⊢ (𝑚 = 𝑛 → if(𝑚 = 1, 𝐴, if(𝑚 = 2, 𝐵, ∅)) = if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅))) |
8 | 7 | cbvmptv 5160 | . 2 ⊢ (𝑚 ∈ ℕ ↦ if(𝑚 = 1, 𝐴, if(𝑚 = 2, 𝐵, ∅))) = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅))) |
9 | 1, 2, 3, 8 | ovnsubadd2lem 42804 | 1 ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 ⊆ wss 3933 ∅c0 4288 ifcif 4463 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 Fincfn 8497 ℝcr 10524 1c1 10526 ≤ cle 10664 ℕcn 11626 2c2 11680 +𝑒 cxad 12493 voln*covoln 42695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cc 9845 ax-ac2 9873 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-acn 9359 df-ac 9530 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 df-prod 15248 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-rest 16684 df-0g 16703 df-topgen 16705 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-bases 21482 df-cmp 21923 df-ovol 23992 df-vol 23993 df-sumge0 42522 df-ovoln 42696 |
This theorem is referenced by: ovnsplit 42807 |
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