![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9919 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | 1 | fconst6 6008 | . . . 4 ⊢ (ℝ × {0}):ℝ⟶ℝ |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → (ℝ × {0}):ℝ⟶ℝ) |
4 | snfi 7923 | . . . . 5 ⊢ {0} ∈ Fin | |
5 | rnxpss 5485 | . . . . 5 ⊢ ran (ℝ × {0}) ⊆ {0} | |
6 | ssfi 8065 | . . . . 5 ⊢ (({0} ∈ Fin ∧ ran (ℝ × {0}) ⊆ {0}) → ran (ℝ × {0}) ∈ Fin) | |
7 | 4, 5, 6 | mp2an 704 | . . . 4 ⊢ ran (ℝ × {0}) ∈ Fin |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → ran (ℝ × {0}) ∈ Fin) |
9 | difss 3699 | . . . . . . 7 ⊢ (ran (ℝ × {0}) ∖ {0}) ⊆ ran (ℝ × {0}) | |
10 | 9, 5 | sstri 3577 | . . . . . 6 ⊢ (ran (ℝ × {0}) ∖ {0}) ⊆ {0} |
11 | 10 | sseli 3564 | . . . . 5 ⊢ (𝑥 ∈ (ran (ℝ × {0}) ∖ {0}) → 𝑥 ∈ {0}) |
12 | 11 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → 𝑥 ∈ {0}) |
13 | eldifn 3695 | . . . . 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 185 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → (◡(ℝ × {0}) “ {𝑥}) ∈ dom vol) |
16 | 12, 14 | pm2.21dd 185 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ (ran (ℝ × {0}) ∖ {0})) → (vol‘(◡(ℝ × {0}) “ {𝑥})) ∈ ℝ) |
17 | 3, 8, 15, 16 | i1fd 23254 | . 2 ⊢ (⊤ → (ℝ × {0}) ∈ dom ∫1) |
18 | 17 | trud 1484 | 1 ⊢ (ℝ × {0}) ∈ dom ∫1 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 ⊤wtru 1476 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 × cxp 5036 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 ⟶wf 5800 ‘cfv 5804 Fincfn 7841 ℝcr 9814 0cc0 9815 volcvol 23039 ∫1citg1 23190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041 df-mbf 23194 df-itg1 23195 |
This theorem is referenced by: itg10 23261 i1fmulc 23276 itg2ge0 23308 itg20 23310 itg2addnclem 32631 itg2addnc 32634 ftc1anclem8 32662 |
Copyright terms: Public domain | W3C validator |