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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-pinftynminfty | Structured version Visualization version GIF version |
Description: The extended complex numbers +∞ and -∞ are different. (Contributed by BJ, 27-Jun-2019.) |
Ref | Expression |
---|---|
bj-pinftynminfty | ⊢ +∞ ≠ -∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pire 26309 | . . . . . . 7 ⊢ π ∈ ℝ | |
2 | pipos 26311 | . . . . . . 7 ⊢ 0 < π | |
3 | 1, 2 | gt0ne0ii 11757 | . . . . . 6 ⊢ π ≠ 0 |
4 | 3 | nesymi 2997 | . . . . 5 ⊢ ¬ 0 = π |
5 | 1 | renegcli 11528 | . . . . . . . 8 ⊢ -π ∈ ℝ |
6 | 5 | rexri 11279 | . . . . . . 7 ⊢ -π ∈ ℝ* |
7 | 0red 11224 | . . . . . . . 8 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → 0 ∈ ℝ) | |
8 | lt0neg2 11728 | . . . . . . . . . . 11 ⊢ (π ∈ ℝ → (0 < π ↔ -π < 0)) | |
9 | 1, 8 | ax-mp 5 | . . . . . . . . . 10 ⊢ (0 < π ↔ -π < 0) |
10 | 2, 9 | mpbi 229 | . . . . . . . . 9 ⊢ -π < 0 |
11 | 10 | a1i 11 | . . . . . . . 8 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → -π < 0) |
12 | 0re 11223 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
13 | 12, 1, 2 | ltleii 11344 | . . . . . . . . 9 ⊢ 0 ≤ π |
14 | 13 | a1i 11 | . . . . . . . 8 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → 0 ≤ π) |
15 | elioc2 13394 | . . . . . . . 8 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → (0 ∈ (-π(,]π) ↔ (0 ∈ ℝ ∧ -π < 0 ∧ 0 ≤ π))) | |
16 | 7, 11, 14, 15 | mpbir3and 1341 | . . . . . . 7 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → 0 ∈ (-π(,]π)) |
17 | 6, 1, 16 | mp2an 689 | . . . . . 6 ⊢ 0 ∈ (-π(,]π) |
18 | simpr 484 | . . . . . . . 8 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → π ∈ ℝ) | |
19 | 5, 12, 1 | lttri 11347 | . . . . . . . . . 10 ⊢ ((-π < 0 ∧ 0 < π) → -π < π) |
20 | 10, 2, 19 | mp2an 689 | . . . . . . . . 9 ⊢ -π < π |
21 | 20 | a1i 11 | . . . . . . . 8 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → -π < π) |
22 | 1 | leidi 11755 | . . . . . . . . 9 ⊢ π ≤ π |
23 | 22 | a1i 11 | . . . . . . . 8 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → π ≤ π) |
24 | elioc2 13394 | . . . . . . . 8 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → (π ∈ (-π(,]π) ↔ (π ∈ ℝ ∧ -π < π ∧ π ≤ π))) | |
25 | 18, 21, 23, 24 | mpbir3and 1341 | . . . . . . 7 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → π ∈ (-π(,]π)) |
26 | 6, 1, 25 | mp2an 689 | . . . . . 6 ⊢ π ∈ (-π(,]π) |
27 | bj-inftyexpiinj 36557 | . . . . . 6 ⊢ ((0 ∈ (-π(,]π) ∧ π ∈ (-π(,]π)) → (0 = π ↔ (+∞ei‘0) = (+∞ei‘π))) | |
28 | 17, 26, 27 | mp2an 689 | . . . . 5 ⊢ (0 = π ↔ (+∞ei‘0) = (+∞ei‘π)) |
29 | 4, 28 | mtbi 322 | . . . 4 ⊢ ¬ (+∞ei‘0) = (+∞ei‘π) |
30 | df-bj-minfty 36572 | . . . . 5 ⊢ -∞ = (+∞ei‘π) | |
31 | 30 | eqeq2i 2744 | . . . 4 ⊢ ((+∞ei‘0) = -∞ ↔ (+∞ei‘0) = (+∞ei‘π)) |
32 | 29, 31 | mtbir 323 | . . 3 ⊢ ¬ (+∞ei‘0) = -∞ |
33 | df-bj-pinfty 36568 | . . . 4 ⊢ +∞ = (+∞ei‘0) | |
34 | 33 | eqeq1i 2736 | . . 3 ⊢ (+∞ = -∞ ↔ (+∞ei‘0) = -∞) |
35 | 32, 34 | mtbir 323 | . 2 ⊢ ¬ +∞ = -∞ |
36 | 35 | neir 2942 | 1 ⊢ +∞ ≠ -∞ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ℝcr 11115 0cc0 11116 ℝ*cxr 11254 < clt 11255 ≤ cle 11256 -cneg 11452 (,]cioc 13332 πcpi 16017 +∞eicinftyexpi 36554 +∞cpinfty 36567 -∞cminfty 36571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ioc 13336 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-fac 14241 df-bc 14270 df-hash 14298 df-shft 15021 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15640 df-ef 16018 df-sin 16020 df-cos 16021 df-pi 16023 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19229 df-cmn 19698 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-fbas 21231 df-fg 21232 df-cnfld 21235 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 df-limc 25716 df-dv 25717 df-bj-inftyexpi 36555 df-bj-pinfty 36568 df-bj-minfty 36572 |
This theorem is referenced by: (None) |
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