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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 596 | 1 ⊢ (𝜑 → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 (class class class)co 7419 ℝcr 11114 + caddc 11118 +𝑒 cxad 13151 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-mulcl 11177 ax-i2m1 11183 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11260 df-mnf 11261 df-xr 11262 df-xadd 13154 |
| This theorem is used by: xpncan 13293 xleadd1a 13295 xadddilem 13336 ismet2 24541 mettri2 24549 prdsxmetlem 24576 bl2in 24608 xblss2ps 24609 methaus 24728 metustexhalf 24764 metdcnlem 25045 metnrmlem3 25070 iscau3 25488 vtxdfiun 29890 vtxdginducedm1fi 29952 infleinflem1 46143 infleinflem2 46144 limsupgtlem 46549 ismbl3 46758 meadjunre 47248 hspmbllem1 47398 hspmbllem2 47399 hspmbllem3 47400 ovolval5lem1 47424 |
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