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Mirrors > Home > ILE Home > Th. List > maxabslemab | GIF version |
Description: Lemma for maxabs 11166. A variation of maxleim 11162- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 21-Dec-2021.) |
Ref | Expression |
---|---|
maxabslemab.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
maxabslemab.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
maxabslemab.ab | ⊢ (𝜑 → 𝐴 < 𝐵) |
Ref | Expression |
---|---|
maxabslemab | ⊢ (𝜑 → (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | maxabslemab.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | 1 | recnd 7941 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
3 | maxabslemab.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 3 | recnd 7941 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
5 | 2, 4, 2 | ppncand 8263 | . . 3 ⊢ (𝜑 → ((𝐵 + 𝐴) + (𝐵 − 𝐴)) = (𝐵 + 𝐵)) |
6 | 4, 2 | addcomd 8063 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
7 | maxabslemab.ab | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵) | |
8 | 3, 1, 7 | ltled 8031 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
9 | 3, 1, 8 | abssuble0d 11134 | . . . 4 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (𝐵 − 𝐴)) |
10 | 6, 9 | oveq12d 5869 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) = ((𝐵 + 𝐴) + (𝐵 − 𝐴))) |
11 | 2 | 2timesd 9113 | . . 3 ⊢ (𝜑 → (2 · 𝐵) = (𝐵 + 𝐵)) |
12 | 5, 10, 11 | 3eqtr4rd 2214 | . 2 ⊢ (𝜑 → (2 · 𝐵) = ((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵)))) |
13 | 4, 2 | addcld 7932 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
14 | 1, 3 | resubcld 8293 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
15 | 9, 14 | eqeltrd 2247 | . . . . 5 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ∈ ℝ) |
16 | 15 | recnd 7941 | . . . 4 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ∈ ℂ) |
17 | 13, 16 | addcld 7932 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) ∈ ℂ) |
18 | 2cnd 8944 | . . 3 ⊢ (𝜑 → 2 ∈ ℂ) | |
19 | 2ap0 8964 | . . . 4 ⊢ 2 # 0 | |
20 | 19 | a1i 9 | . . 3 ⊢ (𝜑 → 2 # 0) |
21 | 17, 18, 2, 20 | divmulapd 8722 | . 2 ⊢ (𝜑 → ((((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) = 𝐵 ↔ (2 · 𝐵) = ((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))))) |
22 | 12, 21 | mpbird 166 | 1 ⊢ (𝜑 → (((𝐴 + 𝐵) + (abs‘(𝐴 − 𝐵))) / 2) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 class class class wbr 3987 ‘cfv 5196 (class class class)co 5851 ℝcr 7766 0cc0 7767 + caddc 7770 · cmul 7772 < clt 7947 − cmin 8083 # cap 8493 / cdiv 8582 2c2 8922 abscabs 10954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-n0 9129 df-z 9206 df-uz 9481 df-seqfrec 10395 df-exp 10469 df-cj 10799 df-re 10800 df-im 10801 df-rsqrt 10955 df-abs 10956 |
This theorem is referenced by: maxabslemlub 11164 |
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