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Mirrors > Home > MPE Home > Th. List > Mathboxes > ex-ovoliunnfl | Structured version Visualization version GIF version |
Description: Demonstration of ovoliunnfl 34815. (Contributed by Brendan Leahy, 21-Nov-2017.) |
Ref | Expression |
---|---|
ex-ovoliunnfl | ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))) | |
2 | eqid 2818 | . . 3 ⊢ (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚))) = (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚))) | |
3 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚)) | |
4 | 3 | sseq1d 3995 | . . . . . . 7 ⊢ (𝑛 = 𝑚 → ((𝑓‘𝑛) ⊆ ℝ ↔ (𝑓‘𝑚) ⊆ ℝ)) |
5 | 2fveq3 6668 | . . . . . . . 8 ⊢ (𝑛 = 𝑚 → (vol*‘(𝑓‘𝑛)) = (vol*‘(𝑓‘𝑚))) | |
6 | 5 | eleq1d 2894 | . . . . . . 7 ⊢ (𝑛 = 𝑚 → ((vol*‘(𝑓‘𝑛)) ∈ ℝ ↔ (vol*‘(𝑓‘𝑚)) ∈ ℝ)) |
7 | 4, 6 | anbi12d 630 | . . . . . 6 ⊢ (𝑛 = 𝑚 → (((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ) ↔ ((𝑓‘𝑚) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑚)) ∈ ℝ))) |
8 | 7 | rspccva 3619 | . . . . 5 ⊢ ((∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑓‘𝑚) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑚)) ∈ ℝ)) |
9 | 8 | simpld 495 | . . . 4 ⊢ ((∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ⊆ ℝ) |
10 | 9 | adantll 710 | . . 3 ⊢ (((𝑓 Fn ℕ ∧ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ)) ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ⊆ ℝ) |
11 | 8 | simprd 496 | . . . 4 ⊢ ((∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑓‘𝑚)) ∈ ℝ) |
12 | 11 | adantll 710 | . . 3 ⊢ (((𝑓 Fn ℕ ∧ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ)) ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑓‘𝑚)) ∈ ℝ) |
13 | 1, 2, 10, 12 | ovoliun 24033 | . 2 ⊢ ((𝑓 Fn ℕ ∧ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ)) → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < )) |
14 | 13 | ovoliunnfl 34815 | 1 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ⊆ wss 3933 ∪ cuni 4830 class class class wbr 5057 ↦ cmpt 5137 Fn wfn 6343 ‘cfv 6348 ≼ cdom 8495 ℝcr 10524 1c1 10526 + caddc 10528 ℕcn 11626 seqcseq 13357 vol*covol 23990 |
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-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 |
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-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-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-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-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-n0 11886 df-z 11970 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-rest 16684 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-bases 21482 df-cmp 21923 df-ovol 23992 |
This theorem is referenced by: (None) |
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