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Theorem xleadd1 13294

Description: Weakened version of xleadd1a 13292 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xleadd1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶)))

**Proof of Theorem xleadd1**
Step	Hyp	Ref	Expression
1		rexr 11305	. . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
2		xleadd1a 13292	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶))
3	2	ex 412	. . 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 13278	. . . . 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 13278	. . . . 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 13252	. . . . 5 ⊢ (𝐶 ∈ ℝ^* → -_𝑒𝐶 ∈ ℝ^*)
13	6, 12	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → -_𝑒𝐶 ∈ ℝ^*)
14		xleadd1a 13292	. . . . 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 412	. . . 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 13290	. . . . 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 13290	. . . . 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 5161	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (((𝐴 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) ≤ ((𝐵 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) ↔ 𝐴 ≤ 𝐵))
22	16, 21	sylibd 239	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶) → 𝐴 ≤ 𝐵))
23	4, 22	impbid 212	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 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 ℝ^*cxr 11292 ≤ cle 11294 -_𝑒cxne 13149 +_𝑒 cxad 13150

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-xneg 13152 df-xadd 13153

This theorem is referenced by: xltadd1 13295 xsubge0 13300 xlesubadd 13302

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