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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 13276. (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 13276 | . 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 11116 + caddc 11120 +𝑒 cxad 13153 |
| 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 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-mulcl 11179 ax-i2m1 11185 |
| 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 11262 df-mnf 11263 df-xr 11264 df-xadd 13156 |
| This theorem is used by: xpncan 13295 xleadd1a 13297 xadddilem 13338 ismet2 24543 mettri2 24551 prdsxmetlem 24578 bl2in 24610 xblss2ps 24611 methaus 24730 metustexhalf 24766 metdcnlem 25047 metnrmlem3 25072 iscau3 25490 vtxdfiun 29892 vtxdginducedm1fi 29954 infleinflem1 46145 infleinflem2 46146 limsupgtlem 46551 ismbl3 46760 meadjunre 47250 hspmbllem1 47400 hspmbllem2 47401 hspmbllem3 47402 ovolval5lem1 47426 |
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