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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > leadd12dd | Structured version Visualization version GIF version |
Description: Addition to both sides of 'less than or equal to'. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
leadd12dd.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
leadd12dd.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
leadd12dd.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
leadd12dd.d | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
leadd12dd.ac | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
leadd12dd.bd | ⊢ (𝜑 → 𝐵 ≤ 𝐷) |
Ref | Expression |
---|---|
leadd12dd | ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leadd12dd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | leadd12dd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 1, 2 | readdcld 11281 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
4 | leadd12dd.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 4, 2 | readdcld 11281 | . 2 ⊢ (𝜑 → (𝐶 + 𝐵) ∈ ℝ) |
6 | leadd12dd.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
7 | 4, 6 | readdcld 11281 | . 2 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ) |
8 | leadd12dd.ac | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐶) | |
9 | 1, 4, 2, 8 | leadd1dd 11866 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐵)) |
10 | leadd12dd.bd | . . 3 ⊢ (𝜑 → 𝐵 ≤ 𝐷) | |
11 | 2, 6, 4, 10 | leadd2dd 11867 | . 2 ⊢ (𝜑 → (𝐶 + 𝐵) ≤ (𝐶 + 𝐷)) |
12 | 3, 5, 7, 9, 11 | letrd 11409 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7426 ℝcr 11145 + caddc 11149 ≤ cle 11287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 |
This theorem is referenced by: sge0xaddlem1 45850 hoidmvlelem2 46013 hspmbllem2 46044 smfmullem1 46208 |
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