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Mirrors > Home > MPE Home > Th. List > Mathboxes > fz0addcom | Structured version Visualization version GIF version |
Description: The addition of two members of a finite set of sequential integers starting at 0 is commutative. (Contributed by Alexander van der Vekens, 22-May-2018.) (Revised by Alexander van der Vekens, 9-Jun-2018.) |
Ref | Expression |
---|---|
fz0addcom | ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfznn0 13294 | . . 3 ⊢ (𝐴 ∈ (0...𝑁) → 𝐴 ∈ ℕ0) | |
2 | 1 | nn0cnd 12241 | . 2 ⊢ (𝐴 ∈ (0...𝑁) → 𝐴 ∈ ℂ) |
3 | elfznn0 13294 | . . 3 ⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℕ0) | |
4 | 3 | nn0cnd 12241 | . 2 ⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℂ) |
5 | addcom 11107 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | |
6 | 2, 4, 5 | syl2an 595 | 1 ⊢ ((𝐴 ∈ (0...𝑁) ∧ 𝐵 ∈ (0...𝑁)) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 (class class class)co 7260 ℂcc 10816 0cc0 10818 + caddc 10821 ...cfz 13184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 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 2708 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 ax-cnex 10874 ax-resscn 10875 ax-1cn 10876 ax-icn 10877 ax-addcl 10878 ax-addrcl 10879 ax-mulcl 10880 ax-mulrcl 10881 ax-mulcom 10882 ax-addass 10883 ax-mulass 10884 ax-distr 10885 ax-i2m1 10886 ax-1ne0 10887 ax-1rid 10888 ax-rnegex 10889 ax-rrecex 10890 ax-cnre 10891 ax-pre-lttri 10892 ax-pre-lttrn 10893 ax-pre-ltadd 10894 ax-pre-mulgt0 10895 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3429 df-sbc 3717 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6259 df-on 6260 df-lim 6261 df-suc 6262 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-f1 6428 df-fo 6429 df-f1o 6430 df-fv 6431 df-riota 7217 df-ov 7263 df-oprab 7264 df-mpo 7265 df-om 7693 df-1st 7809 df-2nd 7810 df-frecs 8073 df-wrecs 8104 df-recs 8178 df-rdg 8217 df-er 8461 df-en 8697 df-dom 8698 df-sdom 8699 df-pnf 10958 df-mnf 10959 df-xr 10960 df-ltxr 10961 df-le 10962 df-sub 11153 df-neg 11154 df-nn 11920 df-n0 12180 df-z 12266 df-uz 12528 df-fz 13185 |
This theorem is referenced by: (None) |
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