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Mirrors > Home > ILE Home > Th. List > rexaddd | GIF version |
Description: The extended real addition operation when both arguments are real. Deduction version of rexadd 9884. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
rexaddd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
rexaddd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
rexaddd | ⊢ (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexaddd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | rexaddd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | rexadd 9884 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) | |
4 | 1, 2, 3 | syl2anc 411 | 1 ⊢ (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 (class class class)co 5897 ℝcr 7841 + caddc 7845 +𝑒 cxad 9802 |
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 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1re 7936 ax-addrcl 7939 ax-rnegex 7951 |
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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-xadd 9805 |
This theorem is referenced by: xpncan 9903 xleadd1a 9905 xltadd1 9908 xleaddadd 9919 xrbdtri 11319 ismet2 14331 xblss2ps 14381 |
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