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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 11900 | . . . 4 ⊢ 0 ∈ ℕ0 | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
5 | 1, 2, 4 | reprval 31991 | . 2 ⊢ (𝜑 → (𝐴(repr‘0)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
6 | fzo0 13056 | . . . . . . . . 9 ⊢ (0..^0) = ∅ | |
7 | 6 | sumeq1i 15047 | . . . . . . . 8 ⊢ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = Σ𝑎 ∈ ∅ (𝑐‘𝑎) |
8 | sum0 15070 | . . . . . . . 8 ⊢ Σ𝑎 ∈ ∅ (𝑐‘𝑎) = 0 | |
9 | 7, 8 | eqtri 2821 | . . . . . . 7 ⊢ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 0 |
10 | 9 | eqeq1i 2803 | . . . . . 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 5175 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
13 | 12 | snid 4561 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
14 | nnex 11631 | . . . . . . . . . . 11 ⊢ ℕ ∈ V | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → ℕ ∈ V) |
16 | 15, 1 | ssexd 5192 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V) |
17 | mapdm0 8404 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐴 ↑m ∅) = {∅}) | |
18 | 16, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑m ∅) = {∅}) |
19 | 13, 18 | eleqtrrid 2897 | . . . . . . 7 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑m ∅)) |
20 | 6 | oveq2i 7146 | . . . . . . 7 ⊢ (𝐴 ↑m (0..^0)) = (𝐴 ↑m ∅) |
21 | 19, 20 | eleqtrrdi 2901 | . . . . . 6 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑m (0..^0))) |
22 | 21 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 0) → ∅ ∈ (𝐴 ↑m (0..^0))) |
23 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑀 = 0) → 𝑀 = 0) | |
24 | 23 | eqcomd 2804 | . . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 0) → 0 = 𝑀) |
25 | 20, 18 | syl5eq 2845 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑m (0..^0)) = {∅}) |
26 | 25 | eleq2d 2875 | . . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑m (0..^0)) ↔ 𝑐 ∈ {∅})) |
27 | 26 | biimpa 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑐 ∈ {∅}) |
28 | elsni 4542 | . . . . . . 7 ⊢ (𝑐 ∈ {∅} → 𝑐 = ∅) | |
29 | 27, 28 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑐 = ∅) |
30 | 29 | ad4ant13 750 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) ∧ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) → 𝑐 = ∅) |
31 | 11, 22, 24, 30 | rabeqsnd 30271 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 = 0) → {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀} = {∅}) |
32 | 31 | eqcomd 2804 | . . 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 768 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → ¬ 𝑀 = 0) | |
35 | 34 | neqned 2994 | . . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑀 ≠ 0) |
36 | 35 | necomd 3042 | . . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 0 ≠ 𝑀) |
37 | 33, 36 | eqnetrd 3054 | . . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) ≠ 𝑀) |
38 | 37 | neneqd 2992 | . . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → ¬ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) |
39 | 38 | ralrimiva 3149 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → ∀𝑐 ∈ (𝐴 ↑m (0..^0)) ¬ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) |
40 | rabeq0 4292 | . . . . 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 237 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀} = ∅) |
42 | 41 | eqcomd 2804 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → ∅ = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
43 | 32, 42 | ifeqda 4460 | . 2 ⊢ (𝜑 → if(𝑀 = 0, {∅}, ∅) = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
44 | 5, 43 | eqtr4d 2836 | 1 ⊢ (𝜑 → (𝐴(repr‘0)𝑀) = if(𝑀 = 0, {∅}, ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 ifcif 4425 {csn 4525 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 0cc0 10526 ℕcn 11625 ℕ0cn0 11885 ℤcz 11969 ..^cfzo 13028 Σcsu 15034 reprcrepr 31989 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-repr 31990 |
This theorem is referenced by: breprexp 32014 |
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