![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xnn0add4d | Structured version Visualization version GIF version |
Description: Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 13278. (Contributed by AV, 12-Dec-2020.) |
Ref | Expression |
---|---|
xnn0add4d.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ0*) |
xnn0add4d.2 | ⊢ (𝜑 → 𝐵 ∈ ℕ0*) |
xnn0add4d.3 | ⊢ (𝜑 → 𝐶 ∈ ℕ0*) |
xnn0add4d.4 | ⊢ (𝜑 → 𝐷 ∈ ℕ0*) |
Ref | Expression |
---|---|
xnn0add4d | ⊢ (𝜑 → ((𝐴 +𝑒 𝐵) +𝑒 (𝐶 +𝑒 𝐷)) = ((𝐴 +𝑒 𝐶) +𝑒 (𝐵 +𝑒 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xnn0add4d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℕ0*) | |
2 | xnn0xrnemnf 12552 | . . 3 ⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
4 | xnn0add4d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℕ0*) | |
5 | xnn0xrnemnf 12552 | . . 3 ⊢ (𝐵 ∈ ℕ0* → (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞)) |
7 | xnn0add4d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℕ0*) | |
8 | xnn0xrnemnf 12552 | . . 3 ⊢ (𝐶 ∈ ℕ0* → (𝐶 ∈ ℝ* ∧ 𝐶 ≠ -∞)) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ* ∧ 𝐶 ≠ -∞)) |
10 | xnn0add4d.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℕ0*) | |
11 | xnn0xrnemnf 12552 | . . 3 ⊢ (𝐷 ∈ ℕ0* → (𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞)) | |
12 | 10, 11 | syl 17 | . 2 ⊢ (𝜑 → (𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞)) |
13 | 3, 6, 9, 12 | xadd4d 13278 | 1 ⊢ (𝜑 → ((𝐴 +𝑒 𝐵) +𝑒 (𝐶 +𝑒 𝐷)) = ((𝐴 +𝑒 𝐶) +𝑒 (𝐵 +𝑒 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 (class class class)co 7404 -∞cmnf 11242 ℝ*cxr 11243 ℕ0*cxnn0 12540 +𝑒 cxad 13086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-nn 12209 df-n0 12469 df-xnn0 12541 df-xadd 13089 |
This theorem is referenced by: vtxdun 28718 vtxdginducedm1 28780 |
Copyright terms: Public domain | W3C validator |