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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 13281. (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 12555 | . . 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 12555 | . . 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 12555 | . . 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 12555 | . . 3 ⊢ (𝐷 ∈ ℕ0* → (𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞)) | |
12 | 10, 11 | syl 17 | . 2 ⊢ (𝜑 → (𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞)) |
13 | 3, 6, 9, 12 | xadd4d 13281 | 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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 (class class class)co 7408 -∞cmnf 11245 ℝ*cxr 11246 ℕ0*cxnn0 12543 +𝑒 cxad 13089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-nn 12212 df-n0 12472 df-xnn0 12544 df-xadd 13092 |
This theorem is referenced by: vtxdun 28735 vtxdginducedm1 28797 |
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