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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnsgrpnmnd | Structured version Visualization version GIF version |
Description: The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.) |
Ref | Expression |
---|---|
nnsgrp.m | ⊢ 𝑀 = (ℂfld ↾s ℕ) |
Ref | Expression |
---|---|
nnsgrpnmnd | ⊢ 𝑀 ∉ Mnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnsscn 12006 | . . . 4 ⊢ ℕ ⊆ ℂ | |
2 | nnsgrp.m | . . . . 5 ⊢ 𝑀 = (ℂfld ↾s ℕ) | |
3 | 2 | cnfldsrngbas 45363 | . . . 4 ⊢ (ℕ ⊆ ℂ → ℕ = (Base‘𝑀)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ ℕ = (Base‘𝑀) |
5 | nnex 12007 | . . . 4 ⊢ ℕ ∈ V | |
6 | 2 | cnfldsrngadd 45364 | . . . 4 ⊢ (ℕ ∈ V → + = (+g‘𝑀)) |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝑀) |
8 | 4, 7 | isnmnd 18417 | . 2 ⊢ (∀𝑧 ∈ ℕ ∃𝑥 ∈ ℕ (𝑧 + 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd) |
9 | 1nn 12012 | . . . 4 ⊢ 1 ∈ ℕ | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑧 ∈ ℕ → 1 ∈ ℕ) |
11 | oveq2 7303 | . . . . 5 ⊢ (𝑥 = 1 → (𝑧 + 𝑥) = (𝑧 + 1)) | |
12 | id 22 | . . . . 5 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
13 | 11, 12 | neeq12d 3000 | . . . 4 ⊢ (𝑥 = 1 → ((𝑧 + 𝑥) ≠ 𝑥 ↔ (𝑧 + 1) ≠ 1)) |
14 | 13 | adantl 481 | . . 3 ⊢ ((𝑧 ∈ ℕ ∧ 𝑥 = 1) → ((𝑧 + 𝑥) ≠ 𝑥 ↔ (𝑧 + 1) ≠ 1)) |
15 | nnne0 12035 | . . . . 5 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0) | |
16 | 15 | necomd 2994 | . . . 4 ⊢ (𝑧 ∈ ℕ → 0 ≠ 𝑧) |
17 | 1cnd 10998 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 1 ∈ ℂ) | |
18 | nncn 12009 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
19 | 17, 17, 18 | subadd2d 11379 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → ((1 − 1) = 𝑧 ↔ (𝑧 + 1) = 1)) |
20 | 1m1e0 12073 | . . . . . . . 8 ⊢ (1 − 1) = 0 | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → (1 − 1) = 0) |
22 | 21 | eqeq1d 2735 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → ((1 − 1) = 𝑧 ↔ 0 = 𝑧)) |
23 | 19, 22 | bitr3d 280 | . . . . 5 ⊢ (𝑧 ∈ ℕ → ((𝑧 + 1) = 1 ↔ 0 = 𝑧)) |
24 | 23 | necon3bid 2983 | . . . 4 ⊢ (𝑧 ∈ ℕ → ((𝑧 + 1) ≠ 1 ↔ 0 ≠ 𝑧)) |
25 | 16, 24 | mpbird 256 | . . 3 ⊢ (𝑧 ∈ ℕ → (𝑧 + 1) ≠ 1) |
26 | 10, 14, 25 | rspcedvd 3565 | . 2 ⊢ (𝑧 ∈ ℕ → ∃𝑥 ∈ ℕ (𝑧 + 𝑥) ≠ 𝑥) |
27 | 8, 26 | mprg 3065 | 1 ⊢ 𝑀 ∉ Mnd |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1537 ∈ wcel 2101 ≠ wne 2938 ∉ wnel 3044 ∃wrex 3068 Vcvv 3434 ⊆ wss 3889 ‘cfv 6447 (class class class)co 7295 ℂcc 10897 0cc0 10899 1c1 10900 + caddc 10902 − cmin 11233 ℕcn 12001 Basecbs 16940 ↾s cress 16969 +gcplusg 16990 Mndcmnd 18413 ℂfldccnfld 20625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-addf 10978 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-fz 13268 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-starv 17005 df-tset 17009 df-ple 17010 df-ds 17012 df-unif 17013 df-mnd 18414 df-cnfld 20626 |
This theorem is referenced by: (None) |
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