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Mirrors > Home > MPE Home > Th. List > ltaddrp2d | Structured version Visualization version GIF version |
Description: Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpgecld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
rpgecld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
Ref | Expression |
---|---|
ltaddrp2d | ⊢ (𝜑 → 𝐴 < (𝐵 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgecld.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | rpgecld.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | 1, 2 | ltaddrpd 12189 | . 2 ⊢ (𝜑 → 𝐴 < (𝐴 + 𝐵)) |
4 | 1 | recnd 10385 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
5 | 2 | rpcnd 12158 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
6 | 4, 5 | addcomd 10557 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
7 | 3, 6 | breqtrd 4899 | 1 ⊢ (𝜑 → 𝐴 < (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 class class class wbr 4873 (class class class)co 6905 ℝcr 10251 + caddc 10255 < clt 10391 ℝ+crp 12112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-ltxr 10396 df-rp 12113 |
This theorem is referenced by: lhop1 24176 cxp2limlem 25115 logdiflbnd 25134 lgamucov 25177 bposlem1 25422 pntpbnd1a 25687 pntibndlem3 25694 pntlemb 25699 pntlemp 25712 2sqmod 30193 madjusmdetlem2 30439 bccolsum 32167 2timesgt 40299 wallispilem4 41079 wallispi 41081 wallispi2lem1 41082 wallispi2lem2 41083 stirlinglem6 41090 stirlinglem7 41091 stirlinglem10 41094 stirlinglem11 41095 dirkertrigeqlem1 41109 fourierdlem42 41160 nnfoctbdjlem 41463 |
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