xle2add - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xle2add		Structured version Visualization version GIF version

Theorem xle2add 13237

Description: Extended real version of le2add 11695. (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 765	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐴 ∈ ℝ^)
2		simprl 769	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐶 ∈ ℝ^)
3		simplr 767	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐵 ∈ ℝ^)
4		xleadd1a 13231	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐶) → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵))
5	4	ex 413	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐶 → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵)))
6	1, 2, 3, 5	syl3anc 1371	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → (𝐴 ≤ 𝐶 → (𝐴 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐵)))
7		simprr 771	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → 𝐷 ∈ ℝ^)
8		xleadd2a 13232	. . . 4 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐵 ≤ 𝐷) → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷))
9	8	ex 413	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐷 → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
10	3, 7, 2, 9	syl3anc 1371	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^*)) → (𝐵 ≤ 𝐷 → (𝐶 +_𝑒 𝐵) ≤ (𝐶 +_𝑒 𝐷)))
11		xaddcl 13217	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
12	11	adantr 481	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
13		xaddcl 13217	. . . 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 13217	. . . 4 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^) → (𝐶 +_𝑒 𝐷) ∈ ℝ^)
16	15	adantl 482	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐷 ∈ ℝ^)) → (𝐶 +_𝑒 𝐷) ∈ ℝ^)
17		xrletr 13136	. . 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 1371	. 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 396 ∧ w3a 1087 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7408 ℝ^*cxr 11246 ≤ cle 11248 +_𝑒 cxad 13089

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-xadd 13092

This theorem is referenced by: metnrmlem3 24376 xraddge02 31964 xrofsup 31975 esumpmono 33072 xadd0ge 44020 sge0split 45115

W3C validator