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Mirrors > Home > MPE Home > Th. List > xleadd1 | Structured version Visualization version GIF version |
Description: Weakened version of xleadd1a 12987 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xleadd1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 11021 | . . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ*) | |
2 | xleadd1a 12987 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶)) | |
3 | 2 | ex 413 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ 𝐵 → (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶))) |
4 | 1, 3 | syl3an3 1164 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 → (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶))) |
5 | simp1 1135 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ*) | |
6 | 1 | 3ad2ant3 1134 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ*) |
7 | xaddcl 12973 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 +𝑒 𝐶) ∈ ℝ*) | |
8 | 5, 6, 7 | syl2anc 584 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 +𝑒 𝐶) ∈ ℝ*) |
9 | simp2 1136 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ*) | |
10 | xaddcl 12973 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐵 +𝑒 𝐶) ∈ ℝ*) | |
11 | 9, 6, 10 | syl2anc 584 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐵 +𝑒 𝐶) ∈ ℝ*) |
12 | xnegcl 12947 | . . . . 5 ⊢ (𝐶 ∈ ℝ* → -𝑒𝐶 ∈ ℝ*) | |
13 | 6, 12 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → -𝑒𝐶 ∈ ℝ*) |
14 | xleadd1a 12987 | . . . . 5 ⊢ ((((𝐴 +𝑒 𝐶) ∈ ℝ* ∧ (𝐵 +𝑒 𝐶) ∈ ℝ* ∧ -𝑒𝐶 ∈ ℝ*) ∧ (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶)) → ((𝐴 +𝑒 𝐶) +𝑒 -𝑒𝐶) ≤ ((𝐵 +𝑒 𝐶) +𝑒 -𝑒𝐶)) | |
15 | 14 | ex 413 | . . . 4 ⊢ (((𝐴 +𝑒 𝐶) ∈ ℝ* ∧ (𝐵 +𝑒 𝐶) ∈ ℝ* ∧ -𝑒𝐶 ∈ ℝ*) → ((𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶) → ((𝐴 +𝑒 𝐶) +𝑒 -𝑒𝐶) ≤ ((𝐵 +𝑒 𝐶) +𝑒 -𝑒𝐶))) |
16 | 8, 11, 13, 15 | syl3anc 1370 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → ((𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶) → ((𝐴 +𝑒 𝐶) +𝑒 -𝑒𝐶) ≤ ((𝐵 +𝑒 𝐶) +𝑒 -𝑒𝐶))) |
17 | xpncan 12985 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → ((𝐴 +𝑒 𝐶) +𝑒 -𝑒𝐶) = 𝐴) | |
18 | 17 | 3adant2 1130 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → ((𝐴 +𝑒 𝐶) +𝑒 -𝑒𝐶) = 𝐴) |
19 | xpncan 12985 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → ((𝐵 +𝑒 𝐶) +𝑒 -𝑒𝐶) = 𝐵) | |
20 | 19 | 3adant1 1129 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → ((𝐵 +𝑒 𝐶) +𝑒 -𝑒𝐶) = 𝐵) |
21 | 18, 20 | breq12d 5087 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (((𝐴 +𝑒 𝐶) +𝑒 -𝑒𝐶) ≤ ((𝐵 +𝑒 𝐶) +𝑒 -𝑒𝐶) ↔ 𝐴 ≤ 𝐵)) |
22 | 16, 21 | sylibd 238 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → ((𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶) → 𝐴 ≤ 𝐵)) |
23 | 4, 22 | impbid 211 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 +𝑒 𝐶) ≤ (𝐵 +𝑒 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℝcr 10870 ℝ*cxr 11008 ≤ cle 11010 -𝑒cxne 12845 +𝑒 cxad 12846 |
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-cnex 10927 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-reu 3072 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-iun 4926 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-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-xneg 12848 df-xadd 12849 |
This theorem is referenced by: xltadd1 12990 xsubge0 12995 xlesubadd 12997 |
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