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Mirrors > Home > ILE Home > Th. List > ltaddsub | GIF version |
Description: 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.) |
Ref | Expression |
---|---|
ltaddsub | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 939 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ) | |
2 | simp3 941 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ) | |
3 | simp2 940 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ) | |
4 | 2, 3 | resubcld 7541 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 − 𝐵) ∈ ℝ) |
5 | ltadd1 7589 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐶 − 𝐵) ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < (𝐶 − 𝐵) ↔ (𝐴 + 𝐵) < ((𝐶 − 𝐵) + 𝐵))) | |
6 | 1, 4, 3, 5 | syl3anc 1170 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < (𝐶 − 𝐵) ↔ (𝐴 + 𝐵) < ((𝐶 − 𝐵) + 𝐵))) |
7 | 2 | recnd 7198 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℂ) |
8 | 3 | recnd 7198 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℂ) |
9 | 7, 8 | npcand 7479 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 − 𝐵) + 𝐵) = 𝐶) |
10 | 9 | breq2d 3799 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < ((𝐶 − 𝐵) + 𝐵) ↔ (𝐴 + 𝐵) < 𝐶)) |
11 | 6, 10 | bitr2d 187 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∧ w3a 920 ∈ wcel 1434 class class class wbr 3787 (class class class)co 5537 ℝcr 7031 + caddc 7035 < clt 7204 − cmin 7335 |
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-sep 3898 ax-pow 3950 ax-pr 3966 ax-un 4190 ax-setind 4282 ax-cnex 7118 ax-resscn 7119 ax-1cn 7120 ax-icn 7122 ax-addcl 7123 ax-addrcl 7124 ax-mulcl 7125 ax-addcom 7127 ax-addass 7129 ax-distr 7131 ax-i2m1 7132 ax-0id 7135 ax-rnegex 7136 ax-cnre 7138 ax-pre-ltadd 7143 |
This theorem depends on definitions: df-bi 115 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-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3386 df-sn 3406 df-pr 3407 df-op 3409 df-uni 3604 df-br 3788 df-opab 3842 df-id 4050 df-xp 4371 df-rel 4372 df-cnv 4373 df-co 4374 df-dm 4375 df-iota 4891 df-fun 4928 df-fv 4934 df-riota 5493 df-ov 5540 df-oprab 5541 df-mpt2 5542 df-pnf 7206 df-mnf 7207 df-ltxr 7209 df-sub 7337 df-neg 7338 |
This theorem is referenced by: ltaddsub2 7597 ltsub13 7603 ltsub2 7619 ltaddsubi 7666 ltaddsubd 7701 iooshf 9040 |
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