xleadd1 - Metamath Proof Explorer

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Theorem xleadd1 13287

Description: Weakened version of xleadd1a 13285 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 11261	. . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
2		xleadd1a 13285	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶))
3	2	ex 417	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐵 → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶)))
4	1, 3	syl3an3 1182	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶)))
5		simp1 1153	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ^*)
6	1	3ad2ant3 1152	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ^*)
7		xaddcl 13271	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐶) ∈ ℝ^)
8	5, 6, 7	syl2anc 595	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 +_𝑒 𝐶) ∈ ℝ^*)
9		simp2 1154	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ^*)
10		xaddcl 13271	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
11	9, 6, 10	syl2anc 595	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐵 +_𝑒 𝐶) ∈ ℝ^*)
12		xnegcl 13245	. . . . 5 ⊢ (𝐶 ∈ ℝ^* → -_𝑒𝐶 ∈ ℝ^*)
13	6, 12	syl 18	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → -_𝑒𝐶 ∈ ℝ^*)
14		xleadd1a 13285	. . . . 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 417	. . . 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 1397	. . 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 13283	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) = 𝐴)
18	17	3adant2 1148	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) = 𝐴)
19		xpncan 13283	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐵 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) = 𝐵)
20	19	3adant1 1147	. . . 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 1102 = wceq 1569 ∈ wcel 2142 class class class wbr 5108 (class class class)co 7412 ℝcr 11105 ℝ^*cxr 11248 ≤ cle 11250 -_𝑒cxne 13140 +_𝑒 cxad 13141

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-po 5568 df-so 5569 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7984 df-2nd 7985 df-er 8692 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-xneg 13143 df-xadd 13144

This theorem is used by: xltadd1 13288 xsubge0 13293 xlesubadd 13295

W3C validator