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Mirrors > Home > ILE Home > Th. List > rexsub | GIF version |
Description: Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
rexsub | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝑒𝐵) = (𝐴 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexneg 9774 | . . . 4 ⊢ (𝐵 ∈ ℝ → -𝑒𝐵 = -𝐵) | |
2 | 1 | adantl 275 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -𝑒𝐵 = -𝐵) |
3 | 2 | oveq2d 5866 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝑒𝐵) = (𝐴 +𝑒 -𝐵)) |
4 | renegcl 8167 | . . 3 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
5 | rexadd 9796 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝐵) = (𝐴 + -𝐵)) | |
6 | 4, 5 | sylan2 284 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝐵) = (𝐴 + -𝐵)) |
7 | recn 7894 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
8 | recn 7894 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
9 | negsub 8154 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
10 | 7, 8, 9 | syl2an 287 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
11 | 3, 6, 10 | 3eqtrd 2207 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 -𝑒𝐵) = (𝐴 − 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 (class class class)co 5850 ℂcc 7759 ℝcr 7760 + caddc 7764 − cmin 8077 -cneg 8078 -𝑒cxne 9713 +𝑒 cxad 9714 |
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-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 |
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-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-sub 8079 df-neg 8080 df-xneg 9716 df-xadd 9717 |
This theorem is referenced by: xposdif 9826 blss2ps 13159 blss2 13160 |
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