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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 11248 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | 1 | fconst6 6787 | . . . 4 ⊢ (ℝ × {0}):ℝ⟶ℝ |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → (ℝ × {0}):ℝ⟶ℝ) |
4 | snfi 9069 | . . . . 5 ⊢ {0} ∈ Fin | |
5 | rnxpss 6178 | . . . . 5 ⊢ ran (ℝ × {0}) ⊆ {0} | |
6 | ssfi 9198 | . . . . 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 4128 | . . . . . . 7 ⊢ (ran (ℝ × {0}) ∖ {0}) ⊆ ran (ℝ × {0}) | |
10 | 9, 5 | sstri 3986 | . . . . . 6 ⊢ (ran (ℝ × {0}) ∖ {0}) ⊆ {0} |
11 | 10 | sseli 3972 | . . . . 5 ⊢ (𝑥 ∈ (ran (ℝ × {0}) ∖ {0}) → 𝑥 ∈ {0}) |
12 | 11 | adantl 480 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → 𝑥 ∈ {0}) |
13 | eldifn 4124 | . . . . 5 ⊢ (𝑥 ∈ (ran (ℝ × {0}) ∖ {0}) → ¬ 𝑥 ∈ {0}) | |
14 | 13 | adantl 480 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → ¬ 𝑥 ∈ {0}) |
15 | 12, 14 | pm2.21dd 194 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → (◡(ℝ × {0}) “ {𝑥}) ∈ dom vol) |
16 | 12, 14 | pm2.21dd 194 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → (vol‘(◡(ℝ × {0}) “ {𝑥})) ∈ ℝ) |
17 | 3, 8, 15, 16 | i1fd 25654 | . 2 ⊢ (⊤ → (ℝ × {0}) ∈ dom ∫1) |
18 | 17 | mptru 1540 | 1 ⊢ (ℝ × {0}) ∈ dom ∫1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 ⊤wtru 1534 ∈ wcel 2098 ∖ cdif 3941 ⊆ wss 3944 {csn 4630 × cxp 5676 ◡ccnv 5677 dom cdm 5678 ran crn 5679 “ cima 5681 ⟶wf 6545 ‘cfv 6549 Fincfn 8964 ℝcr 11139 0cc0 11140 volcvol 25436 ∫1citg1 25588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-oi 9535 df-dju 9926 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-xadd 13128 df-ioo 13363 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-sum 15669 df-xmet 21289 df-met 21290 df-ovol 25437 df-vol 25438 df-mbf 25592 df-itg1 25593 |
This theorem is referenced by: itg10 25661 i1fmulc 25677 itg2ge0 25709 itg20 25711 itg2addnclem 37275 itg2addnc 37278 ftc1anclem8 37304 |
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