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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 12221 | . . . 4 ⊢ ℕ ⊆ ℂ | |
2 | nnsgrp.m | . . . . 5 ⊢ 𝑀 = (ℂfld ↾s ℕ) | |
3 | 2 | cnfldsrngbas 47116 | . . . 4 ⊢ (ℕ ⊆ ℂ → ℕ = (Base‘𝑀)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ ℕ = (Base‘𝑀) |
5 | nnex 12222 | . . . 4 ⊢ ℕ ∈ V | |
6 | 2 | cnfldsrngadd 47117 | . . . 4 ⊢ (ℕ ∈ V → + = (+g‘𝑀)) |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝑀) |
8 | 4, 7 | isnmnd 18671 | . 2 ⊢ (∀𝑧 ∈ ℕ ∃𝑥 ∈ ℕ (𝑧 + 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd) |
9 | 1nn 12227 | . . . 4 ⊢ 1 ∈ ℕ | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑧 ∈ ℕ → 1 ∈ ℕ) |
11 | oveq2 7413 | . . . . 5 ⊢ (𝑥 = 1 → (𝑧 + 𝑥) = (𝑧 + 1)) | |
12 | id 22 | . . . . 5 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
13 | 11, 12 | neeq12d 2996 | . . . 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 12250 | . . . . 5 ⊢ (𝑧 ∈ ℕ → 𝑧 ≠ 0) | |
16 | 15 | necomd 2990 | . . . 4 ⊢ (𝑧 ∈ ℕ → 0 ≠ 𝑧) |
17 | 1cnd 11213 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 1 ∈ ℂ) | |
18 | nncn 12224 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
19 | 17, 17, 18 | subadd2d 11594 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → ((1 − 1) = 𝑧 ↔ (𝑧 + 1) = 1)) |
20 | 1m1e0 12288 | . . . . . . . 8 ⊢ (1 − 1) = 0 | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝑧 ∈ ℕ → (1 − 1) = 0) |
22 | 21 | eqeq1d 2728 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → ((1 − 1) = 𝑧 ↔ 0 = 𝑧)) |
23 | 19, 22 | bitr3d 281 | . . . . 5 ⊢ (𝑧 ∈ ℕ → ((𝑧 + 1) = 1 ↔ 0 = 𝑧)) |
24 | 23 | necon3bid 2979 | . . . 4 ⊢ (𝑧 ∈ ℕ → ((𝑧 + 1) ≠ 1 ↔ 0 ≠ 𝑧)) |
25 | 16, 24 | mpbird 257 | . . 3 ⊢ (𝑧 ∈ ℕ → (𝑧 + 1) ≠ 1) |
26 | 10, 14, 25 | rspcedvd 3608 | . 2 ⊢ (𝑧 ∈ ℕ → ∃𝑥 ∈ ℕ (𝑧 + 𝑥) ≠ 𝑥) |
27 | 8, 26 | mprg 3061 | 1 ⊢ 𝑀 ∉ Mnd |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∉ wnel 3040 ∃wrex 3064 Vcvv 3468 ⊆ wss 3943 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 − cmin 11448 ℕcn 12216 Basecbs 17153 ↾s cress 17182 +gcplusg 17206 Mndcmnd 18667 ℂfldccnfld 21240 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-mnd 18668 df-cnfld 21241 |
This theorem is referenced by: (None) |
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