xltadd1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xltadd1		Structured version Visualization version GIF version

Theorem xltadd1 12643

Description: Extended real version of ltadd1 11101. (Contributed by Mario Carneiro, 23-Aug-2015.)

**Assertion**
Ref	Expression
xltadd1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 +_𝑒 𝐶) < (𝐵 +_𝑒 𝐶)))

**Proof of Theorem xltadd1**
Step	Hyp	Ref	Expression
1		xleadd1 12642	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ (𝐵 +_𝑒 𝐶) ≤ (𝐴 +_𝑒 𝐶)))
2	1	3com12 1119	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐵 ≤ 𝐴 ↔ (𝐵 +_𝑒 𝐶) ≤ (𝐴 +_𝑒 𝐶)))
3	2	notbid 320	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (¬ 𝐵 ≤ 𝐴 ↔ ¬ (𝐵 +_𝑒 𝐶) ≤ (𝐴 +_𝑒 𝐶)))
4		xrltnle 10702	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
5	4	3adant3 1128	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
6		simp1 1132	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ^*)
7		rexr 10681	. . . . 5 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
8	7	3ad2ant3 1131	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ^*)
9		xaddcl 12626	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐴 +_𝑒 𝐶) ∈ ℝ^)
10	6, 8, 9	syl2anc 586	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 +_𝑒 𝐶) ∈ ℝ^*)
11		simp2 1133	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ^*)
12		xaddcl 12626	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 +_𝑒 𝐶) ∈ ℝ^)
13	11, 8, 12	syl2anc 586	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐵 +_𝑒 𝐶) ∈ ℝ^*)
14		xrltnle 10702	. . 3 ⊢ (((𝐴 +_𝑒 𝐶) ∈ ℝ^* ∧ (𝐵 +_𝑒 𝐶) ∈ ℝ^*) → ((𝐴 +_𝑒 𝐶) < (𝐵 +_𝑒 𝐶) ↔ ¬ (𝐵 +_𝑒 𝐶) ≤ (𝐴 +_𝑒 𝐶)))
15	10, 13, 14	syl2anc 586	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → ((𝐴 +_𝑒 𝐶) < (𝐵 +_𝑒 𝐶) ↔ ¬ (𝐵 +_𝑒 𝐶) ≤ (𝐴 +_𝑒 𝐶)))
16	3, 5, 15	3bitr4d 313	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 +_𝑒 𝐶) < (𝐵 +_𝑒 𝐶)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 +_𝑒 cxad 12499

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-xneg 12501 df-xadd 12502

This theorem is referenced by: xltadd2 12644 xlt2add 12647 hashunsnggt 13749

W3C validator