xleadd1 - Metamath Proof Explorer

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Theorem xleadd1 13392

Description: Weakened version of xleadd1a 13390 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 11362	. . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
2		xleadd1a 13390	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +_e 𝐶) ≤ (𝐵 +_e 𝐶))
3	2	ex 418	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐵 → (𝐴 +_e 𝐶) ≤ (𝐵 +_e 𝐶)))
4	1, 3	syl3an3 1183	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 → (𝐴 +_e 𝐶) ≤ (𝐵 +_e 𝐶)))
5		simp1 1154	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ^*)
6	1	3ad2ant3 1153	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ^*)
7		xaddcl 13376	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_e 𝐶) ∈ ℝ^)
8	5, 6, 7	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 +_e 𝐶) ∈ ℝ^*)
9		simp2 1155	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ^*)
10		xaddcl 13376	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 +_e 𝐶) ∈ ℝ^)
11	9, 6, 10	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐵 +_e 𝐶) ∈ ℝ^*)
12		xnegcl 13350	. . . . 5 ⊢ (𝐶 ∈ ℝ^* → -_e𝐶 ∈ ℝ^*)
13	6, 12	syl 18	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → -_e𝐶 ∈ ℝ^*)
14		xleadd1a 13390	. . . . 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 418	. . . 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 1398	. . 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 13388	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_e 𝐶) +_e -_e𝐶) = 𝐴)
18	17	3adant2 1149	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_e 𝐶) +_e -_e𝐶) = 𝐴)
19		xpncan 13388	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐵 +_e 𝐶) +_e -_e𝐶) = 𝐵)
20	19	3adant1 1148	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐵 +_e 𝐶) +_e -_e𝐶) = 𝐵)
21	18, 20	breq12d 5115	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (((𝐴 +_e 𝐶) +_e -_e𝐶) ≤ ((𝐵 +_e 𝐶) +_e -_e𝐶) ↔ 𝐴 ≤ 𝐵))
22	16, 21	sylibd 242	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_e 𝐶) ≤ (𝐵 +_e 𝐶) → 𝐴 ≤ 𝐵))
23	4, 22	impbid 215	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 +_e 𝐶) ≤ (𝐵 +_e 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 class class class wbr 5102 (class class class)co 7424 ℝcr 11206 ℝ^*cxr 11349 ≤ cle 11351 -_ecxne 13245 +_e cxad 13246

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8005 df-2nd 8006 df-er 8725 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-xneg 13248 df-xadd 13249

This theorem is used by: xltadd1 13393 xsubge0 13398 xlesubadd 13400

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