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Mirrors > Home > MPE Home > Th. List > rexadd | Structured version Visualization version GIF version |
Description: The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
rexadd | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 11284 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 11284 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xaddval 13228 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))) | |
4 | 1, 2, 3 | syl2an 595 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))) |
5 | renepnf 11286 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
6 | ifnefalse 4536 | . . . . 5 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) |
8 | renemnf 11287 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
9 | ifnefalse 4536 | . . . . 5 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) |
11 | 7, 10 | eqtrd 2767 | . . 3 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) |
12 | renepnf 11286 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞) | |
13 | ifnefalse 4536 | . . . . 5 ⊢ (𝐵 ≠ +∞ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) |
15 | renemnf 11287 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞) | |
16 | ifnefalse 4536 | . . . . 5 ⊢ (𝐵 ≠ -∞ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝐵 ∈ ℝ → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) = (𝐴 + 𝐵)) |
18 | 14, 17 | eqtrd 2767 | . . 3 ⊢ (𝐵 ∈ ℝ → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) = (𝐴 + 𝐵)) |
19 | 11, 18 | sylan9eq 2787 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) = (𝐴 + 𝐵)) |
20 | 4, 19 | eqtrd 2767 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ifcif 4524 (class class class)co 7414 ℝcr 11131 0cc0 11132 + caddc 11135 +∞cpnf 11269 -∞cmnf 11270 ℝ*cxr 11271 +𝑒 cxad 13116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-mulcl 11194 ax-i2m1 11200 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-xadd 13119 |
This theorem is referenced by: rexsub 13238 rexaddd 13239 xnn0xaddcl 13240 xaddnemnf 13241 xaddnepnf 13242 xnegid 13243 xaddcom 13245 xaddrid 13246 xnn0xadd0 13252 xnegdi 13253 xaddass 13254 xadddilem 13299 x2times 13304 hashunx 14371 hashunsnggt 14379 isxmet2d 24226 xmeter 24332 vtxdgfival 29276 1loopgrvd2 29310 vdegp1bi 29344 xlt2addrd 32522 xrsmulgzz 32730 xrge0slmod 33054 xrge0iifhom 33528 esumfsupre 33680 esumpfinvallem 33683 omssubadd 33910 probun 34029 heicant 37117 cntotbnd 37258 heiborlem6 37278 supxrgelem 44691 supxrge 44692 infrpge 44705 xrlexaddrp 44706 ovolsplit 45348 sge0tsms 45740 sge0pr 45754 sge0resplit 45766 sge0split 45769 sge0iunmptlemfi 45773 sge0iunmptlemre 45775 sge0xaddlem1 45793 sge0xaddlem2 45794 carageniuncllem1 45881 carageniuncllem2 45882 hoidmv1lelem2 45952 hoidmvlelem2 45956 hspmbllem3 45988 |
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