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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 13076 | . 2 ⊢ (𝜑 → 𝐴 < (𝐴 + 𝐵)) |
4 | 1 | recnd 11267 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
5 | 2 | rpcnd 13045 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
6 | 4, 5 | addcomd 11441 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
7 | 3, 6 | breqtrd 5169 | 1 ⊢ (𝜑 → 𝐴 < (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 class class class wbr 5143 (class class class)co 7415 ℝcr 11132 + caddc 11136 < clt 11273 ℝ+crp 13001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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-ov 7418 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-ltxr 11278 df-rp 13002 |
This theorem is referenced by: lhop1 25941 cxp2limlem 26902 logdiflbnd 26921 lgamucov 26964 bposlem1 27211 2sqmod 27363 pntpbnd1a 27512 pntibndlem3 27519 pntlemb 27524 pntlemp 27537 madjusmdetlem2 33424 bccolsum 35328 2timesgt 44661 wallispilem4 45447 wallispi 45449 wallispi2lem1 45450 wallispi2lem2 45451 stirlinglem6 45458 stirlinglem7 45459 stirlinglem10 45462 stirlinglem11 45463 dirkertrigeqlem1 45477 fourierdlem42 45528 nnfoctbdjlem 45834 |
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