Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12805 | . 2 ⊢ (𝜑 → 𝐴 < (𝐴 + 𝐵)) |
4 | 1 | recnd 11003 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
5 | 2 | rpcnd 12774 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
6 | 4, 5 | addcomd 11177 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
7 | 3, 6 | breqtrd 5100 | 1 ⊢ (𝜑 → 𝐴 < (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℝcr 10870 + caddc 10874 < clt 11009 ℝ+crp 12730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-ltxr 11014 df-rp 12731 |
This theorem is referenced by: lhop1 25178 cxp2limlem 26125 logdiflbnd 26144 lgamucov 26187 bposlem1 26432 2sqmod 26584 pntpbnd1a 26733 pntibndlem3 26740 pntlemb 26745 pntlemp 26758 madjusmdetlem2 31778 bccolsum 33705 2timesgt 42827 wallispilem4 43609 wallispi 43611 wallispi2lem1 43612 wallispi2lem2 43613 stirlinglem6 43620 stirlinglem7 43621 stirlinglem10 43624 stirlinglem11 43625 dirkertrigeqlem1 43639 fourierdlem42 43690 nnfoctbdjlem 43993 |
Copyright terms: Public domain | W3C validator |