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Theorem xle2add 13153

Description: Extended real version of le2add 11594. (Contributed by Mario Carneiro, 23-Aug-2015.)

**Assertion**
Ref	Expression
xle2add	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))

**Proof of Theorem xle2add**
Step	Hyp	Ref	Expression
1		simpll 766	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐴 ∈ ℝ^)
2		simprl 770	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐶 ∈ ℝ^)
3		simplr 768	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐵 ∈ ℝ^)
4		xleadd1a 13147	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐶) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵))
5	4	ex 412	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐶 → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵)))
6	1, 2, 3, 5	syl3anc 1373	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → (𝐴 ≤ 𝐶 → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵)))
7		simprr 772	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐷 ∈ ℝ^)
8		xleadd2a 13148	. . . 4 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐵 ≤ 𝐷) → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷))
9	8	ex 412	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐷 → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
10	3, 7, 2, 9	syl3anc 1373	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → (𝐵 ≤ 𝐷 → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
11		xaddcl 13133	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
12	11	adantr 480	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
13		xaddcl 13133	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 +_𝑒 𝐵) ∈ ℝ^)
14	2, 3, 13	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (𝐶 +_𝑒 𝐵) ∈ ℝ^)
15		xaddcl 13133	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝐶 +_𝑒 𝐷) ∈ ℝ^)
16	15	adantl 481	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (𝐶 +_𝑒 𝐷) ∈ ℝ^)
17		xrletr 13052	. . 3 ⊢ (((𝐴 +_𝑒 𝐵) ∈ ℝ^* ∧ (𝐶 +_𝑒 𝐵) ∈ ℝ^* ∧ (𝐶 +_𝑒 𝐷) ∈ ℝ^*) → (((𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵) ∧ (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
18	12, 14, 16, 17	syl3anc 1373	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → (((𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵) ∧ (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
19	6, 10, 18	syl2and 608	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2111 class class class wbr 5086 (class class class)co 7341 ℝ^*cxr 11140 ≤ cle 11142 +_𝑒 cxad 13004

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5506 df-po 5519 df-so 5520 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-ov 7344 df-oprab 7345 df-mpo 7346 df-1st 7916 df-2nd 7917 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-xadd 13007

This theorem is referenced by: metnrmlem3 24772 xraddge02 32732 xrofsup 32742 esumpmono 34084 xadd0ge 45360 sge0split 46447

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