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Mirrors > Home > ILE Home > Th. List > xnpcan | GIF version |
Description: Extended real version of npcan 8230. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnpcan | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 8067 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
2 | xnegneg 9902 | . . . . 5 ⊢ (𝐵 ∈ ℝ* → -𝑒-𝑒𝐵 = 𝐵) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝐵 ∈ ℝ → -𝑒-𝑒𝐵 = 𝐵) |
4 | 3 | adantl 277 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → -𝑒-𝑒𝐵 = 𝐵) |
5 | 4 | oveq2d 5935 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 -𝑒-𝑒𝐵) = ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵)) |
6 | rexneg 9899 | . . . 4 ⊢ (𝐵 ∈ ℝ → -𝑒𝐵 = -𝐵) | |
7 | renegcl 8282 | . . . 4 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
8 | 6, 7 | eqeltrd 2270 | . . 3 ⊢ (𝐵 ∈ ℝ → -𝑒𝐵 ∈ ℝ) |
9 | xpncan 9940 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ -𝑒𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 -𝑒-𝑒𝐵) = 𝐴) | |
10 | 8, 9 | sylan2 286 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 -𝑒-𝑒𝐵) = 𝐴) |
11 | 5, 10 | eqtr3d 2228 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 (class class class)co 5919 ℝcr 7873 ℝ*cxr 8055 -cneg 8193 -𝑒cxne 9838 +𝑒 cxad 9839 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-distr 7978 ax-i2m1 7979 ax-0id 7982 ax-rnegex 7983 ax-cnre 7985 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-pnf 8058 df-mnf 8059 df-xr 8060 df-sub 8194 df-neg 8195 df-xneg 9841 df-xadd 9842 |
This theorem is referenced by: xsubge0 9950 xlesubadd 9952 xrmaxaddlem 11406 xblss2ps 14583 xblss2 14584 |
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