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Mirrors > Home > ILE Home > Th. List > leaddsub | GIF version |
Description: 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.) |
Ref | Expression |
---|---|
leaddsub | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 998 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ) | |
2 | simp3 1000 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ) | |
3 | simp2 999 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ) | |
4 | 2, 3 | resubcld 8351 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 − 𝐵) ∈ ℝ) |
5 | leadd1 8400 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐶 − 𝐵) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ (𝐶 − 𝐵) ↔ (𝐴 + 𝐵) ≤ ((𝐶 − 𝐵) + 𝐵))) | |
6 | 1, 4, 3, 5 | syl3anc 1248 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ (𝐶 − 𝐵) ↔ (𝐴 + 𝐵) ≤ ((𝐶 − 𝐵) + 𝐵))) |
7 | 2 | recnd 7999 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℂ) |
8 | 3 | recnd 7999 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℂ) |
9 | 7, 8 | npcand 8285 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 − 𝐵) + 𝐵) = 𝐶) |
10 | 9 | breq2d 4027 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ ((𝐶 − 𝐵) + 𝐵) ↔ (𝐴 + 𝐵) ≤ 𝐶)) |
11 | 6, 10 | bitr2d 189 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 − 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 979 ∈ wcel 2158 class class class wbr 4015 (class class class)co 5888 ℝcr 7823 + caddc 7827 ≤ cle 8006 − cmin 8141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 |
This theorem is referenced by: leaddsub2 8409 lesub 8411 lesub2 8427 subge0 8445 div4p1lem1div2 9185 eluzp1m1 9564 eluzsubi 9568 eluzsub 9570 fzen 10056 fznatpl1 10089 seq3f1olemqsumkj 10511 bcval5 10756 uzwodc 12051 hashdvds 12234 |
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