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Mirrors > Home > ILE Home > Th. List > xnn0add4d | GIF version |
Description: Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 9887. (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 9253 | . . 3 ⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
4 | xnn0add4d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℕ0*) | |
5 | xnn0xrnemnf 9253 | . . 3 ⊢ (𝐵 ∈ ℕ0* → (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞)) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞)) |
7 | xnn0add4d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℕ0*) | |
8 | xnn0xrnemnf 9253 | . . 3 ⊢ (𝐶 ∈ ℕ0* → (𝐶 ∈ ℝ* ∧ 𝐶 ≠ -∞)) | |
9 | 7, 8 | syl 14 | . 2 ⊢ (𝜑 → (𝐶 ∈ ℝ* ∧ 𝐶 ≠ -∞)) |
10 | xnn0add4d.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℕ0*) | |
11 | xnn0xrnemnf 9253 | . . 3 ⊢ (𝐷 ∈ ℕ0* → (𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞)) | |
12 | 10, 11 | syl 14 | . 2 ⊢ (𝜑 → (𝐷 ∈ ℝ* ∧ 𝐷 ≠ -∞)) |
13 | 3, 6, 9, 12 | xadd4d 9887 | 1 ⊢ (𝜑 → ((𝐴 +𝑒 𝐵) +𝑒 (𝐶 +𝑒 𝐷)) = ((𝐴 +𝑒 𝐶) +𝑒 (𝐵 +𝑒 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 (class class class)co 5877 -∞cmnf 7992 ℝ*cxr 7993 ℕ0*cxnn0 9241 +𝑒 cxad 9772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1re 7907 ax-addrcl 7910 ax-addcom 7913 ax-addass 7915 ax-rnegex 7922 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-pnf 7996 df-mnf 7997 df-xr 7998 df-inn 8922 df-n0 9179 df-xnn0 9242 df-xadd 9775 |
This theorem is referenced by: (None) |
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