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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 11631 | . . . 4 ⊢ ℕ ⊆ ℂ | |
2 | nnsgrp.m | . . . . 5 ⊢ 𝑀 = (ℂfld ↾s ℕ) | |
3 | 2 | cnfldsrngbas 43913 | . . . 4 ⊢ (ℕ ⊆ ℂ → ℕ = (Base‘𝑀)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ ℕ = (Base‘𝑀) |
5 | nnex 11632 | . . . 4 ⊢ ℕ ∈ V | |
6 | 2 | cnfldsrngadd 43914 | . . . 4 ⊢ (ℕ ∈ V → + = (+g‘𝑀)) |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝑀) |
8 | 4, 7 | isnmnd 17903 | . 2 ⊢ (∀𝑧 ∈ ℕ ∃𝑥 ∈ ℕ (𝑧 + 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd) |
9 | 1nn 11637 | . . . 4 ⊢ 1 ∈ ℕ | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑧 ∈ ℕ → 1 ∈ ℕ) |
11 | oveq2 7153 | . . . . 5 ⊢ (𝑥 = 1 → (𝑧 + 𝑥) = (𝑧 + 1)) | |
12 | id 22 | . . . . 5 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
13 | 11, 12 | neeq12d 3074 | . . . 4 ⊢ (𝑥 = 1 → ((𝑧 + 𝑥) ≠ 𝑥 ↔ (𝑧 + 1) ≠ 1)) |
14 | 13 | adantl 482 | . . 3 ⊢ ((𝑧 ∈ ℕ ∧ 𝑥 = 1) → ((𝑧 + 𝑥) ≠ 𝑥 ↔ (𝑧 + 1) ≠ 1)) |
15 | nnne0 11659 | . . . . 5 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0) | |
16 | 15 | necomd 3068 | . . . 4 ⊢ (𝑧 ∈ ℕ → 0 ≠ 𝑧) |
17 | 1cnd 10624 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 1 ∈ ℂ) | |
18 | nncn 11634 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
19 | 17, 17, 18 | subadd2d 11004 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → ((1 − 1) = 𝑧 ↔ (𝑧 + 1) = 1)) |
20 | 1m1e0 11697 | . . . . . . . 8 ⊢ (1 − 1) = 0 | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → (1 − 1) = 0) |
22 | 21 | eqeq1d 2820 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → ((1 − 1) = 𝑧 ↔ 0 = 𝑧)) |
23 | 19, 22 | bitr3d 282 | . . . . 5 ⊢ (𝑧 ∈ ℕ → ((𝑧 + 1) = 1 ↔ 0 = 𝑧)) |
24 | 23 | necon3bid 3057 | . . . 4 ⊢ (𝑧 ∈ ℕ → ((𝑧 + 1) ≠ 1 ↔ 0 ≠ 𝑧)) |
25 | 16, 24 | mpbird 258 | . . 3 ⊢ (𝑧 ∈ ℕ → (𝑧 + 1) ≠ 1) |
26 | 10, 14, 25 | rspcedvd 3623 | . 2 ⊢ (𝑧 ∈ ℕ → ∃𝑥 ∈ ℕ (𝑧 + 𝑥) ≠ 𝑥) |
27 | 8, 26 | mprg 3149 | 1 ⊢ 𝑀 ∉ Mnd |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∉ wnel 3120 ∃wrex 3136 Vcvv 3492 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 − cmin 10858 ℕcn 11626 Basecbs 16471 ↾s cress 16472 +gcplusg 16553 Mndcmnd 17899 ℂfldccnfld 20473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-mnd 17900 df-cnfld 20474 |
This theorem is referenced by: (None) |
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