xraddge02 - Mathbox for Thierry Arnoux

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Theorem xraddge02 32959

Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.)

**Assertion**
Ref	Expression
xraddge02	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ 𝐵 → 𝐴 ≤ (𝐴 +_𝑒 𝐵)))

**Proof of Theorem xraddge02**
Step	Hyp	Ref	Expression
1		xrleid 13153	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
2	1	adantr 484	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐴 ≤ 𝐴)
3		simpl 486	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		0xr 11229	. . . . 5 ⊢ 0 ∈ ℝ^*
5	3, 4	jctir 528	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ∈ ℝ^ ∧ 0 ∈ ℝ^*))
6		xle2add 13262	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ∈ ℝ^) ∧ (𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*)) → ((𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 +_𝑒 0) ≤ (𝐴 +_𝑒 𝐵)))
7	5, 6	mpancom 698	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 +_𝑒 0) ≤ (𝐴 +_𝑒 𝐵)))
8	2, 7	mpand 705	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ 𝐵 → (𝐴 +_𝑒 0) ≤ (𝐴 +_𝑒 𝐵)))
9		xaddrid 13244	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)
10	9	breq1d 5110	. . 3 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 +_𝑒 0) ≤ (𝐴 +_𝑒 𝐵) ↔ 𝐴 ≤ (𝐴 +_𝑒 𝐵)))
11	10	adantr 484	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 0) ≤ (𝐴 +_𝑒 𝐵) ↔ 𝐴 ≤ (𝐴 +_𝑒 𝐵)))
12	8, 11	sylibd 241	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ 𝐵 → 𝐴 ≤ (𝐴 +_𝑒 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∈ wcel 2142 class class class wbr 5100 (class class class)co 7396 0cc0 11073 ℝ^*cxr 11215 ≤ cle 11217 +_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-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-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-xadd 13115

This theorem is referenced by: xrge0addge 32960 esummono 34351 esumle 34355 gsumesum 34356 esumlef 34359 measssd 34512 measunl 34513 carsgsigalem 34612

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