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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 11260 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | 1 | fconst6 6798 | . . . 4 ⊢ (ℝ × {0}):ℝ⟶ℝ |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → (ℝ × {0}):ℝ⟶ℝ) |
4 | snfi 9081 | . . . . 5 ⊢ {0} ∈ Fin | |
5 | rnxpss 6193 | . . . . 5 ⊢ ran (ℝ × {0}) ⊆ {0} | |
6 | ssfi 9211 | . . . . 5 ⊢ (({0} ∈ Fin ∧ ran (ℝ × {0}) ⊆ {0}) → ran (ℝ × {0}) ∈ Fin) | |
7 | 4, 5, 6 | mp2an 692 | . . . 4 ⊢ ran (ℝ × {0}) ∈ Fin |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → ran (ℝ × {0}) ∈ Fin) |
9 | difss 4145 | . . . . . . 7 ⊢ (ran (ℝ × {0}) ∖ {0}) ⊆ ran (ℝ × {0}) | |
10 | 9, 5 | sstri 4004 | . . . . . 6 ⊢ (ran (ℝ × {0}) ∖ {0}) ⊆ {0} |
11 | 10 | sseli 3990 | . . . . 5 ⊢ (𝑥 ∈ (ran (ℝ × {0}) ∖ {0}) → 𝑥 ∈ {0}) |
12 | 11 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → 𝑥 ∈ {0}) |
13 | eldifn 4141 | . . . . 5 ⊢ (𝑥 ∈ (ran (ℝ × {0}) ∖ {0}) → ¬ 𝑥 ∈ {0}) | |
14 | 13 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → ¬ 𝑥 ∈ {0}) |
15 | 12, 14 | pm2.21dd 195 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → (◡(ℝ × {0}) “ {𝑥}) ∈ dom vol) |
16 | 12, 14 | pm2.21dd 195 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → (vol‘(◡(ℝ × {0}) “ {𝑥})) ∈ ℝ) |
17 | 3, 8, 15, 16 | i1fd 25729 | . 2 ⊢ (⊤ → (ℝ × {0}) ∈ dom ∫1) |
18 | 17 | mptru 1543 | 1 ⊢ (ℝ × {0}) ∈ dom ∫1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ⊤wtru 1537 ∈ wcel 2105 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 × cxp 5686 ◡ccnv 5687 dom cdm 5688 ran crn 5689 “ cima 5691 ⟶wf 6558 ‘cfv 6562 Fincfn 8983 ℝcr 11151 0cc0 11152 volcvol 25511 ∫1citg1 25663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-xadd 13152 df-ioo 13387 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 df-xmet 21374 df-met 21375 df-ovol 25512 df-vol 25513 df-mbf 25667 df-itg1 25668 |
This theorem is referenced by: itg10 25736 i1fmulc 25752 itg2ge0 25784 itg20 25786 itg2addnclem 37657 itg2addnc 37660 ftc1anclem8 37686 |
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