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| Mirrors > Home > MPE Home > Th. List > rexaddd | Structured version Visualization version GIF version | ||
| Description: The extended real addition operation when both arguments are real. Deduction version of rexadd 13274. (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 13274 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℝcr 11154 + caddc 11158 +𝑒 cxad 13152 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-xadd 13155 |
| This theorem is referenced by: xpncan 13293 xleadd1a 13295 xadddilem 13336 ismet2 24343 mettri2 24351 prdsxmetlem 24378 bl2in 24410 xblss2ps 24411 methaus 24533 metustexhalf 24569 metdcnlem 24858 metnrmlem3 24883 iscau3 25312 vtxdfiun 29500 vtxdginducedm1fi 29562 infleinflem1 45381 infleinflem2 45382 limsupgtlem 45792 ismbl3 46001 meadjunre 46491 hspmbllem1 46641 hspmbllem2 46642 hspmbllem3 46643 ovolval5lem1 46667 |
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