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Mirrors > Home > ILE Home > Th. List > absrpclap | GIF version |
Description: The absolute value of a number apart from zero is a positive real. (Contributed by Jim Kingdon, 11-Aug-2021.) |
Ref | Expression |
---|---|
absrpclap | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (abs‘𝐴) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absval 10025 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
2 | 1 | adantr 270 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
3 | simpl 107 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → 𝐴 ∈ ℂ) | |
4 | 3 | cjmulrcld 9976 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (∗‘𝐴)) ∈ ℝ) |
5 | 3 | cjmulge0d 9978 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → 0 ≤ (𝐴 · (∗‘𝐴))) |
6 | 3 | cjcld 9965 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (∗‘𝐴) ∈ ℂ) |
7 | simpr 108 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → 𝐴 # 0) | |
8 | 3, 7 | cjap0d 9973 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (∗‘𝐴) # 0) |
9 | 3, 6, 7, 8 | mulap0d 7815 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (∗‘𝐴)) # 0) |
10 | 4, 5, 9 | ap0gt0d 7806 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → 0 < (𝐴 · (∗‘𝐴))) |
11 | 4, 10 | elrpd 8852 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴 · (∗‘𝐴)) ∈ ℝ+) |
12 | rpsqrtcl 10065 | . . 3 ⊢ ((𝐴 · (∗‘𝐴)) ∈ ℝ+ → (√‘(𝐴 · (∗‘𝐴))) ∈ ℝ+) | |
13 | 11, 12 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (√‘(𝐴 · (∗‘𝐴))) ∈ ℝ+) |
14 | 2, 13 | eqeltrd 2156 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (abs‘𝐴) ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 class class class wbr 3793 ‘cfv 4932 (class class class)co 5543 ℂcc 7041 0cc0 7043 · cmul 7048 # cap 7748 ℝ+crp 8815 ∗ccj 9864 √csqrt 10020 abscabs 10021 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-coll 3901 ax-sep 3904 ax-nul 3912 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-iinf 4337 ax-cnex 7129 ax-resscn 7130 ax-1cn 7131 ax-1re 7132 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-mulrcl 7137 ax-addcom 7138 ax-mulcom 7139 ax-addass 7140 ax-mulass 7141 ax-distr 7142 ax-i2m1 7143 ax-0lt1 7144 ax-1rid 7145 ax-0id 7146 ax-rnegex 7147 ax-precex 7148 ax-cnre 7149 ax-pre-ltirr 7150 ax-pre-ltwlin 7151 ax-pre-lttrn 7152 ax-pre-apti 7153 ax-pre-ltadd 7154 ax-pre-mulgt0 7155 ax-pre-mulext 7156 ax-arch 7157 ax-caucvg 7158 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rmo 2357 df-rab 2358 df-v 2604 df-sbc 2817 df-csb 2910 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-if 3360 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-iun 3688 df-br 3794 df-opab 3848 df-mpt 3849 df-tr 3884 df-id 4056 df-po 4059 df-iso 4060 df-iord 4129 df-on 4131 df-ilim 4132 df-suc 4134 df-iom 4340 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-1st 5798 df-2nd 5799 df-recs 5954 df-frec 6040 df-pnf 7217 df-mnf 7218 df-xr 7219 df-ltxr 7220 df-le 7221 df-sub 7348 df-neg 7349 df-reap 7742 df-ap 7749 df-div 7828 df-inn 8107 df-2 8165 df-3 8166 df-4 8167 df-n0 8356 df-z 8433 df-uz 8701 df-rp 8816 df-iseq 9522 df-iexp 9573 df-cj 9867 df-re 9868 df-im 9869 df-rsqrt 10022 df-abs 10023 |
This theorem is referenced by: abs00ap 10086 absdivap 10094 absrpclapd 10212 |
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