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 11915 | . . . 4 ⊢ 0 ∈ ℕ0 | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
5 | 1, 2, 4 | reprval 31883 | . 2 ⊢ (𝜑 → (𝐴(repr‘0)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
6 | fzo0 13064 | . . . . . . . . 9 ⊢ (0..^0) = ∅ | |
7 | 6 | sumeq1i 15057 | . . . . . . . 8 ⊢ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = Σ𝑎 ∈ ∅ (𝑐‘𝑎) |
8 | sum0 15080 | . . . . . . . 8 ⊢ Σ𝑎 ∈ ∅ (𝑐‘𝑎) = 0 | |
9 | 7, 8 | eqtri 2846 | . . . . . . 7 ⊢ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 0 |
10 | 9 | eqeq1i 2828 | . . . . . 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 5213 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
13 | 12 | snid 4603 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
14 | nnex 11646 | . . . . . . . . . . 11 ⊢ ℕ ∈ V | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → ℕ ∈ V) |
16 | 15, 1 | ssexd 5230 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V) |
17 | mapdm0 8423 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐴 ↑m ∅) = {∅}) | |
18 | 16, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑m ∅) = {∅}) |
19 | 13, 18 | eleqtrrid 2922 | . . . . . . 7 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑m ∅)) |
20 | 6 | oveq2i 7169 | . . . . . . 7 ⊢ (𝐴 ↑m (0..^0)) = (𝐴 ↑m ∅) |
21 | 19, 20 | eleqtrrdi 2926 | . . . . . 6 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑m (0..^0))) |
22 | 21 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 0) → ∅ ∈ (𝐴 ↑m (0..^0))) |
23 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑀 = 0) → 𝑀 = 0) | |
24 | 23 | eqcomd 2829 | . . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 0) → 0 = 𝑀) |
25 | 20, 18 | syl5eq 2870 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑m (0..^0)) = {∅}) |
26 | 25 | eleq2d 2900 | . . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑m (0..^0)) ↔ 𝑐 ∈ {∅})) |
27 | 26 | biimpa 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑐 ∈ {∅}) |
28 | elsni 4586 | . . . . . . 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 30266 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 = 0) → {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀} = {∅}) |
32 | 31 | eqcomd 2829 | . . 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 3025 | . . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑀 ≠ 0) |
36 | 35 | necomd 3073 | . . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 0 ≠ 𝑀) |
37 | 33, 36 | eqnetrd 3085 | . . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) ≠ 𝑀) |
38 | 37 | neneqd 3023 | . . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → ¬ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) |
39 | 38 | ralrimiva 3184 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → ∀𝑐 ∈ (𝐴 ↑m (0..^0)) ¬ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) |
40 | rabeq0 4340 | . . . . 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 236 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀} = ∅) |
42 | 41 | eqcomd 2829 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → ∅ = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
43 | 32, 42 | ifeqda 4504 | . 2 ⊢ (𝜑 → if(𝑀 = 0, {∅}, ∅) = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
44 | 5, 43 | eqtr4d 2861 | 1 ⊢ (𝜑 → (𝐴(repr‘0)𝑀) = if(𝑀 = 0, {∅}, ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 ifcif 4469 {csn 4569 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 0cc0 10539 ℕcn 11640 ℕ0cn0 11900 ℤcz 11984 ..^cfzo 13036 Σcsu 15044 reprcrepr 31881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-repr 31882 |
This theorem is referenced by: breprexp 31906 |
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