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Mirrors > Home > MPE Home > Th. List > xle2add | Structured version Visualization version GIF version |
Description: Extended real version of le2add 11697. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
xle2add | ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 764 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → 𝐴 ∈ ℝ*) | |
2 | simprl 768 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → 𝐶 ∈ ℝ*) | |
3 | simplr 766 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → 𝐵 ∈ ℝ*) | |
4 | xleadd1a 13235 | . . . 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 1368 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → (𝐴 ≤ 𝐶 → (𝐴 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐵))) |
7 | simprr 770 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → 𝐷 ∈ ℝ*) | |
8 | xleadd2a 13236 | . . . 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 1368 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → (𝐵 ≤ 𝐷 → (𝐶 +𝑒 𝐵) ≤ (𝐶 +𝑒 𝐷))) |
11 | xaddcl 13221 | . . . 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 13221 | . . . 4 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 +𝑒 𝐵) ∈ ℝ*) | |
14 | 2, 3, 13 | syl2anc 583 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → (𝐶 +𝑒 𝐵) ∈ ℝ*) |
15 | xaddcl 13221 | . . . 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 13140 | . . 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 1368 | . 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 607 | 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 1084 ∈ wcel 2098 class class class wbr 5141 (class class class)co 7404 ℝ*cxr 11248 ≤ cle 11250 +𝑒 cxad 13093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-xadd 13096 |
This theorem is referenced by: metnrmlem3 24727 xraddge02 32473 xrofsup 32484 esumpmono 33606 xadd0ge 44584 sge0split 45679 |
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