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Mirrors > Home > ILE Home > Th. List > difgtsumgt | GIF version |
Description: If the difference of a real number and a nonnegative integer is greater than another real number, the sum of the real number and the nonnegative integer is also greater than the other real number. (Contributed by AV, 13-Aug-2021.) |
Ref | Expression |
---|---|
difgtsumgt | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐶 < (𝐴 − 𝐵) → 𝐶 < (𝐴 + 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 7877 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | nn0cn 9115 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℂ) | |
3 | 1, 2 | anim12i 336 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0) → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) |
4 | 3 | 3adant3 1006 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) |
5 | negsub 8137 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
6 | 4, 5 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
7 | 6 | eqcomd 2170 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐴 − 𝐵) = (𝐴 + -𝐵)) |
8 | 7 | breq2d 3988 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐶 < (𝐴 − 𝐵) ↔ 𝐶 < (𝐴 + -𝐵))) |
9 | simp3 988 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ) | |
10 | simp1 986 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ) | |
11 | nn0re 9114 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ) | |
12 | 11 | renegcld 8269 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0 → -𝐵 ∈ ℝ) |
13 | 12 | 3ad2ant2 1008 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → -𝐵 ∈ ℝ) |
14 | 10, 13 | readdcld 7919 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐴 + -𝐵) ∈ ℝ) |
15 | 11 | 3ad2ant2 1008 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ) |
16 | 10, 15 | readdcld 7919 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) |
17 | 9, 14, 16 | 3jca 1166 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ ℝ ∧ (𝐴 + -𝐵) ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ)) |
18 | nn0negleid 9250 | . . . . 5 ⊢ (𝐵 ∈ ℕ0 → -𝐵 ≤ 𝐵) | |
19 | 18 | 3ad2ant2 1008 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → -𝐵 ≤ 𝐵) |
20 | 13, 15, 10, 19 | leadd2dd 8449 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐴 + -𝐵) ≤ (𝐴 + 𝐵)) |
21 | 17, 20 | lelttrdi 8315 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐶 < (𝐴 + -𝐵) → 𝐶 < (𝐴 + 𝐵))) |
22 | 8, 21 | sylbid 149 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ0 ∧ 𝐶 ∈ ℝ) → (𝐶 < (𝐴 − 𝐵) → 𝐶 < (𝐴 + 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 (class class class)co 5836 ℂcc 7742 ℝcr 7743 + caddc 7747 < clt 7924 ≤ cle 7925 − cmin 8060 -cneg 8061 ℕ0cn0 9105 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 |
This theorem is referenced by: difsqpwdvds 12246 |
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