| 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 12541 | . . . 4 ⊢ 0 ∈ ℕ0 | |
| 4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
| 5 | 1, 2, 4 | reprval 34625 | . 2 ⊢ (𝜑 → (𝐴(repr‘0)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
| 6 | fzo0 13723 | . . . . . . . . 9 ⊢ (0..^0) = ∅ | |
| 7 | 6 | sumeq1i 15733 | . . . . . . . 8 ⊢ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = Σ𝑎 ∈ ∅ (𝑐‘𝑎) |
| 8 | sum0 15757 | . . . . . . . 8 ⊢ Σ𝑎 ∈ ∅ (𝑐‘𝑎) = 0 | |
| 9 | 7, 8 | eqtri 2765 | . . . . . . 7 ⊢ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 0 |
| 10 | 9 | eqeq1i 2742 | . . . . . 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 5307 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
| 13 | 12 | snid 4662 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
| 14 | nnex 12272 | . . . . . . . . . . 11 ⊢ ℕ ∈ V | |
| 15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → ℕ ∈ V) |
| 16 | 15, 1 | ssexd 5324 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V) |
| 17 | mapdm0 8882 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐴 ↑m ∅) = {∅}) | |
| 18 | 16, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑m ∅) = {∅}) |
| 19 | 13, 18 | eleqtrrid 2848 | . . . . . . 7 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑m ∅)) |
| 20 | 6 | oveq2i 7442 | . . . . . . 7 ⊢ (𝐴 ↑m (0..^0)) = (𝐴 ↑m ∅) |
| 21 | 19, 20 | eleqtrrdi 2852 | . . . . . 6 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑m (0..^0))) |
| 22 | 21 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 0) → ∅ ∈ (𝐴 ↑m (0..^0))) |
| 23 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑀 = 0) → 𝑀 = 0) | |
| 24 | 23 | eqcomd 2743 | . . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 0) → 0 = 𝑀) |
| 25 | 20, 18 | eqtrid 2789 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑m (0..^0)) = {∅}) |
| 26 | 25 | eleq2d 2827 | . . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑m (0..^0)) ↔ 𝑐 ∈ {∅})) |
| 27 | 26 | biimpa 476 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑐 ∈ {∅}) |
| 28 | elsni 4643 | . . . . . . 7 ⊢ (𝑐 ∈ {∅} → 𝑐 = ∅) | |
| 29 | 27, 28 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑐 = ∅) |
| 30 | 29 | ad4ant13 751 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) ∧ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) → 𝑐 = ∅) |
| 31 | 11, 22, 24, 30 | rabeqsnd 4669 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 = 0) → {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀} = {∅}) |
| 32 | 31 | eqcomd 2743 | . . 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 769 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → ¬ 𝑀 = 0) | |
| 35 | 34 | neqned 2947 | . . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑀 ≠ 0) |
| 36 | 35 | necomd 2996 | . . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 0 ≠ 𝑀) |
| 37 | 33, 36 | eqnetrd 3008 | . . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) ≠ 𝑀) |
| 38 | 37 | neneqd 2945 | . . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → ¬ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) |
| 39 | 38 | ralrimiva 3146 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → ∀𝑐 ∈ (𝐴 ↑m (0..^0)) ¬ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) |
| 40 | rabeq0 4388 | . . . . 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 234 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀} = ∅) |
| 42 | 41 | eqcomd 2743 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → ∅ = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
| 43 | 32, 42 | ifeqda 4562 | . 2 ⊢ (𝜑 → if(𝑀 = 0, {∅}, ∅) = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
| 44 | 5, 43 | eqtr4d 2780 | 1 ⊢ (𝜑 → (𝐴(repr‘0)𝑀) = if(𝑀 = 0, {∅}, ∅)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ifcif 4525 {csn 4626 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 0cc0 11155 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 ..^cfzo 13694 Σcsu 15722 reprcrepr 34623 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-repr 34624 |
| This theorem is referenced by: breprexp 34648 |
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