xleadd1 - Metamath Proof Explorer

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Theorem xleadd1 13425

Description: Weakened version of xleadd1a 13423 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 11394	. . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
2		xleadd1a 13423	. . . 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 13409	. . . . 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 13409	. . . . 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 13383	. . . . 5 ⊢ (𝐶 ∈ ℝ^* → -_e𝐶 ∈ ℝ^*)
13	6, 12	syl 18	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → -_e𝐶 ∈ ℝ^*)
14		xleadd1a 13423	. . . . 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 1399	. . 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 13421	. . . . 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 13421	. . . . 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 5121	. . 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 1570 ∈ wcel 2147 class class class wbr 5108 (class class class)co 7451 ℝcr 11238 ℝ^*cxr 11381 ≤ cle 11383 -_ecxne 13278 +_e cxad 13279

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-po 5566 df-so 5567 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-1st 8035 df-2nd 8036 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-xneg 13281 df-xadd 13282

This theorem is used by: xltadd1 13426 xsubge0 13431 xlesubadd 13433

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