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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 12559 | . 2 ⊢ (𝜑 → 𝐴 < (𝐴 + 𝐵)) |
4 | 1 | recnd 10759 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
5 | 2 | rpcnd 12528 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
6 | 4, 5 | addcomd 10932 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
7 | 3, 6 | breqtrd 5066 | 1 ⊢ (𝜑 → 𝐴 < (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5040 (class class class)co 7182 ℝcr 10626 + caddc 10630 < clt 10765 ℝ+crp 12484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7185 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-ltxr 10770 df-rp 12485 |
This theorem is referenced by: lhop1 24778 cxp2limlem 25725 logdiflbnd 25744 lgamucov 25787 bposlem1 26032 2sqmod 26184 pntpbnd1a 26333 pntibndlem3 26340 pntlemb 26345 pntlemp 26358 madjusmdetlem2 31362 bccolsum 33290 2timesgt 42404 wallispilem4 43191 wallispi 43193 wallispi2lem1 43194 wallispi2lem2 43195 stirlinglem6 43202 stirlinglem7 43203 stirlinglem10 43206 stirlinglem11 43207 dirkertrigeqlem1 43221 fourierdlem42 43272 nnfoctbdjlem 43575 |
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