xleadd1 - Metamath Proof Explorer

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Theorem xleadd1 13222

Description: Weakened version of xleadd1a 13220 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 11227	. . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
2		xleadd1a 13220	. . . 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 1165	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 → (𝐴 +_𝑒 𝐶) ≤ (𝐵 +_𝑒 𝐶)))
5		simp1 1136	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ^*)
6	1	3ad2ant3 1135	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ^*)
7		xaddcl 13206	. . . . 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 1137	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ^*)
10		xaddcl 13206	. . . . 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 13180	. . . . 5 ⊢ (𝐶 ∈ ℝ^* → -_𝑒𝐶 ∈ ℝ^*)
13	6, 12	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → -_𝑒𝐶 ∈ ℝ^*)
14		xleadd1a 13220	. . . . 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 1373	. . 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 13218	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) = 𝐴)
18	17	3adant2 1131	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) = 𝐴)
19		xpncan 13218	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐵 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) = 𝐵)
20	19	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐵 +_𝑒 𝐶) +_𝑒 -_𝑒𝐶) = 𝐵)
21	18, 20	breq12d 5123	. . 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 1540 ∈ wcel 2109 class class class wbr 5110 (class class class)co 7390 ℝcr 11074 ℝ^*cxr 11214 ≤ cle 11216 -_𝑒cxne 13076 +_𝑒 cxad 13077

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-xneg 13079 df-xadd 13080

This theorem is referenced by: xltadd1 13223 xsubge0 13228 xlesubadd 13230

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