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Theorem xleadd1 13258

Description: Weakened version of xleadd1a 13256 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 11228	. . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
2		xleadd1a 13256	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶))
3	2	ex 416	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 ≤ 𝐵 → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶)))
4	1, 3	syl3an3 1178	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶)))
5		simp1 1149	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ^*)
6	1	3ad2ant3 1148	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ^*)
7		xaddcl 13242	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐶) ∈ ℝ^)
8	5, 6, 7	syl2anc 593	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 +_𝑒 𝐶) ∈ ℝ^*)
9		simp2 1150	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ^*)
10		xaddcl 13242	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
11	9, 6, 10	syl2anc 593	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐵 +_𝑒 𝐶) ∈ ℝ^*)
12		xnegcl 13216	. . . . 5 ⊢ (𝐶 ∈ ℝ^* → -_𝑒𝐶 ∈ ℝ^*)
13	6, 12	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → -_𝑒𝐶 ∈ ℝ^*)
14		xleadd1a 13256	. . . . 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 416	. . . 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 1390	. . 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 13254	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) = 𝐴)
18	17	3adant2 1144	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) = 𝐴)
19		xpncan 13254	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐵 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) = 𝐵)
20	19	3adant1 1143	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐵 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) = 𝐵)
21	18, 20	breq12d 5113	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (((𝐴 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) ≤ ((𝐵 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) ↔ 𝐴 ≤ 𝐵))
22	16, 21	sylibd 241	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶) → 𝐴 ≤ 𝐵))
23	4, 22	impbid 214	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 208 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 class class class wbr 5100 (class class class)co 7396 ℝcr 11072 ℝ^*cxr 11215 ≤ cle 11217 -_𝑒cxne 13111 +_𝑒 cxad 13112

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-po 5555 df-so 5556 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7970 df-2nd 7971 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-xneg 13114 df-xadd 13115

This theorem is referenced by: xltadd1 13259 xsubge0 13264 xlesubadd 13266

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