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Mirrors > Home > MPE Home > Th. List > i1f0 | Structured version Visualization version GIF version |
Description: The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
i1f0 | ⊢ (ℝ × {0}) ∈ dom ∫1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10635 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | 1 | fconst6 6562 | . . . 4 ⊢ (ℝ × {0}):ℝ⟶ℝ |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → (ℝ × {0}):ℝ⟶ℝ) |
4 | snfi 8586 | . . . . 5 ⊢ {0} ∈ Fin | |
5 | rnxpss 6022 | . . . . 5 ⊢ ran (ℝ × {0}) ⊆ {0} | |
6 | ssfi 8730 | . . . . 5 ⊢ (({0} ∈ Fin ∧ ran (ℝ × {0}) ⊆ {0}) → ran (ℝ × {0}) ∈ Fin) | |
7 | 4, 5, 6 | mp2an 690 | . . . 4 ⊢ ran (ℝ × {0}) ∈ Fin |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → ran (ℝ × {0}) ∈ Fin) |
9 | difss 4106 | . . . . . . 7 ⊢ (ran (ℝ × {0}) ∖ {0}) ⊆ ran (ℝ × {0}) | |
10 | 9, 5 | sstri 3974 | . . . . . 6 ⊢ (ran (ℝ × {0}) ∖ {0}) ⊆ {0} |
11 | 10 | sseli 3961 | . . . . 5 ⊢ (𝑥 ∈ (ran (ℝ × {0}) ∖ {0}) → 𝑥 ∈ {0}) |
12 | 11 | adantl 484 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → 𝑥 ∈ {0}) |
13 | eldifn 4102 | . . . . 5 ⊢ (𝑥 ∈ (ran (ℝ × {0}) ∖ {0}) → ¬ 𝑥 ∈ {0}) | |
14 | 13 | adantl 484 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → ¬ 𝑥 ∈ {0}) |
15 | 12, 14 | pm2.21dd 197 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → (◡(ℝ × {0}) “ {𝑥}) ∈ dom vol) |
16 | 12, 14 | pm2.21dd 197 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → (vol‘(◡(ℝ × {0}) “ {𝑥})) ∈ ℝ) |
17 | 3, 8, 15, 16 | i1fd 24274 | . 2 ⊢ (⊤ → (ℝ × {0}) ∈ dom ∫1) |
18 | 17 | mptru 1538 | 1 ⊢ (ℝ × {0}) ∈ dom ∫1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ⊤wtru 1532 ∈ wcel 2108 ∖ cdif 3931 ⊆ wss 3934 {csn 4559 × cxp 5546 ◡ccnv 5547 dom cdm 5548 ran crn 5549 “ cima 5551 ⟶wf 6344 ‘cfv 6348 Fincfn 8501 ℝcr 10528 0cc0 10529 volcvol 24056 ∫1citg1 24208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-inf 8899 df-oi 8966 df-dju 9322 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-q 12341 df-rp 12382 df-xadd 12500 df-ioo 12734 df-ico 12736 df-icc 12737 df-fz 12885 df-fzo 13026 df-fl 13154 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-xmet 20530 df-met 20531 df-ovol 24057 df-vol 24058 df-mbf 24212 df-itg1 24213 |
This theorem is referenced by: itg10 24281 i1fmulc 24296 itg2ge0 24328 itg20 24330 itg2addnclem 34935 itg2addnc 34938 ftc1anclem8 34966 |
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