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Mathbox for Alexander van der Vekens |
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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 12245 | . . . 4 ⊢ ℕ ⊆ ℂ | |
2 | nnsgrp.m | . . . . 5 ⊢ 𝑀 = (ℂfld ↾s ℕ) | |
3 | 2 | cnfldsrngbas 47334 | . . . 4 ⊢ (ℕ ⊆ ℂ → ℕ = (Base‘𝑀)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ ℕ = (Base‘𝑀) |
5 | nnex 12246 | . . . 4 ⊢ ℕ ∈ V | |
6 | 2 | cnfldsrngadd 47335 | . . . 4 ⊢ (ℕ ∈ V → + = (+g‘𝑀)) |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝑀) |
8 | 4, 7 | isnmnd 18695 | . 2 ⊢ (∀𝑧 ∈ ℕ ∃𝑥 ∈ ℕ (𝑧 + 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd) |
9 | 1nn 12251 | . . . 4 ⊢ 1 ∈ ℕ | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑧 ∈ ℕ → 1 ∈ ℕ) |
11 | oveq2 7423 | . . . . 5 ⊢ (𝑥 = 1 → (𝑧 + 𝑥) = (𝑧 + 1)) | |
12 | id 22 | . . . . 5 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
13 | 11, 12 | neeq12d 2992 | . . . 4 ⊢ (𝑥 = 1 → ((𝑧 + 𝑥) ≠ 𝑥 ↔ (𝑧 + 1) ≠ 1)) |
14 | 13 | adantl 480 | . . 3 ⊢ ((𝑧 ∈ ℕ ∧ 𝑥 = 1) → ((𝑧 + 𝑥) ≠ 𝑥 ↔ (𝑧 + 1) ≠ 1)) |
15 | nnne0 12274 | . . . . 5 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0) | |
16 | 15 | necomd 2986 | . . . 4 ⊢ (𝑧 ∈ ℕ → 0 ≠ 𝑧) |
17 | 1cnd 11237 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 1 ∈ ℂ) | |
18 | nncn 12248 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
19 | 17, 17, 18 | subadd2d 11618 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → ((1 − 1) = 𝑧 ↔ (𝑧 + 1) = 1)) |
20 | 1m1e0 12312 | . . . . . . . 8 ⊢ (1 − 1) = 0 | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → (1 − 1) = 0) |
22 | 21 | eqeq1d 2727 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → ((1 − 1) = 𝑧 ↔ 0 = 𝑧)) |
23 | 19, 22 | bitr3d 280 | . . . . 5 ⊢ (𝑧 ∈ ℕ → ((𝑧 + 1) = 1 ↔ 0 = 𝑧)) |
24 | 23 | necon3bid 2975 | . . . 4 ⊢ (𝑧 ∈ ℕ → ((𝑧 + 1) ≠ 1 ↔ 0 ≠ 𝑧)) |
25 | 16, 24 | mpbird 256 | . . 3 ⊢ (𝑧 ∈ ℕ → (𝑧 + 1) ≠ 1) |
26 | 10, 14, 25 | rspcedvd 3604 | . 2 ⊢ (𝑧 ∈ ℕ → ∃𝑥 ∈ ℕ (𝑧 + 𝑥) ≠ 𝑥) |
27 | 8, 26 | mprg 3057 | 1 ⊢ 𝑀 ∉ Mnd |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∉ wnel 3036 ∃wrex 3060 Vcvv 3463 ⊆ wss 3940 ‘cfv 6542 (class class class)co 7415 ℂcc 11134 0cc0 11136 1c1 11137 + caddc 11139 − cmin 11472 ℕcn 12240 Basecbs 17177 ↾s cress 17206 +gcplusg 17230 Mndcmnd 18691 ℂfldccnfld 21281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-addf 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-mnd 18692 df-cnfld 21282 |
This theorem is referenced by: (None) |
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