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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > repr0 | Structured version Visualization version GIF version |
Description: There is exactly one representation with no elements (an empty sum), only for 𝑀 = 0. (Contributed by Thierry Arnoux, 2-Dec-2021.) |
Ref | Expression |
---|---|
reprval.a | ⊢ (𝜑 → 𝐴 ⊆ ℕ) |
reprval.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
reprval.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) |
Ref | Expression |
---|---|
repr0 | ⊢ (𝜑 → (𝐴(repr‘0)𝑀) = if(𝑀 = 0, {∅}, ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reprval.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
2 | reprval.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | 0nn0 12428 | . . . 4 ⊢ 0 ∈ ℕ0 | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
5 | 1, 2, 4 | reprval 33223 | . 2 ⊢ (𝜑 → (𝐴(repr‘0)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
6 | fzo0 13596 | . . . . . . . . 9 ⊢ (0..^0) = ∅ | |
7 | 6 | sumeq1i 15583 | . . . . . . . 8 ⊢ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = Σ𝑎 ∈ ∅ (𝑐‘𝑎) |
8 | sum0 15606 | . . . . . . . 8 ⊢ Σ𝑎 ∈ ∅ (𝑐‘𝑎) = 0 | |
9 | 7, 8 | eqtri 2764 | . . . . . . 7 ⊢ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 0 |
10 | 9 | eqeq1i 2741 | . . . . . 6 ⊢ (Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀 ↔ 0 = 𝑀) |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝑐 = ∅ → (Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀 ↔ 0 = 𝑀)) |
12 | 0ex 5264 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
13 | 12 | snid 4622 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
14 | nnex 12159 | . . . . . . . . . . 11 ⊢ ℕ ∈ V | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → ℕ ∈ V) |
16 | 15, 1 | ssexd 5281 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V) |
17 | mapdm0 8780 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐴 ↑m ∅) = {∅}) | |
18 | 16, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑m ∅) = {∅}) |
19 | 13, 18 | eleqtrrid 2845 | . . . . . . 7 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑m ∅)) |
20 | 6 | oveq2i 7368 | . . . . . . 7 ⊢ (𝐴 ↑m (0..^0)) = (𝐴 ↑m ∅) |
21 | 19, 20 | eleqtrrdi 2849 | . . . . . 6 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑m (0..^0))) |
22 | 21 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 0) → ∅ ∈ (𝐴 ↑m (0..^0))) |
23 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑀 = 0) → 𝑀 = 0) | |
24 | 23 | eqcomd 2742 | . . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 0) → 0 = 𝑀) |
25 | 20, 18 | eqtrid 2788 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑m (0..^0)) = {∅}) |
26 | 25 | eleq2d 2823 | . . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑m (0..^0)) ↔ 𝑐 ∈ {∅})) |
27 | 26 | biimpa 477 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑐 ∈ {∅}) |
28 | elsni 4603 | . . . . . . 7 ⊢ (𝑐 ∈ {∅} → 𝑐 = ∅) | |
29 | 27, 28 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑐 = ∅) |
30 | 29 | ad4ant13 749 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) ∧ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) → 𝑐 = ∅) |
31 | 11, 22, 24, 30 | rabeqsnd 31429 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 = 0) → {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀} = {∅}) |
32 | 31 | eqcomd 2742 | . . 3 ⊢ ((𝜑 ∧ 𝑀 = 0) → {∅} = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
33 | 9 | a1i 11 | . . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 0) |
34 | simplr 767 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → ¬ 𝑀 = 0) | |
35 | 34 | neqned 2950 | . . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑀 ≠ 0) |
36 | 35 | necomd 2999 | . . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 0 ≠ 𝑀) |
37 | 33, 36 | eqnetrd 3011 | . . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) ≠ 𝑀) |
38 | 37 | neneqd 2948 | . . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → ¬ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) |
39 | 38 | ralrimiva 3143 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → ∀𝑐 ∈ (𝐴 ↑m (0..^0)) ¬ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) |
40 | rabeq0 4344 | . . . . 5 ⊢ ({𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀} = ∅ ↔ ∀𝑐 ∈ (𝐴 ↑m (0..^0)) ¬ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) | |
41 | 39, 40 | sylibr 233 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀} = ∅) |
42 | 41 | eqcomd 2742 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → ∅ = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
43 | 32, 42 | ifeqda 4522 | . 2 ⊢ (𝜑 → if(𝑀 = 0, {∅}, ∅) = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
44 | 5, 43 | eqtr4d 2779 | 1 ⊢ (𝜑 → (𝐴(repr‘0)𝑀) = if(𝑀 = 0, {∅}, ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 {crab 3407 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 ifcif 4486 {csn 4586 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 0cc0 11051 ℕcn 12153 ℕ0cn0 12413 ℤcz 12499 ..^cfzo 13567 Σcsu 15570 reprcrepr 33221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-repr 33222 |
This theorem is referenced by: breprexp 33246 |
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