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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 12539 | . . . 4 ⊢ 0 ∈ ℕ0 | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℕ0) |
5 | 1, 2, 4 | reprval 34604 | . 2 ⊢ (𝜑 → (𝐴(repr‘0)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
6 | fzo0 13720 | . . . . . . . . 9 ⊢ (0..^0) = ∅ | |
7 | 6 | sumeq1i 15730 | . . . . . . . 8 ⊢ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = Σ𝑎 ∈ ∅ (𝑐‘𝑎) |
8 | sum0 15754 | . . . . . . . 8 ⊢ Σ𝑎 ∈ ∅ (𝑐‘𝑎) = 0 | |
9 | 7, 8 | eqtri 2763 | . . . . . . 7 ⊢ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 0 |
10 | 9 | eqeq1i 2740 | . . . . . 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 5313 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
13 | 12 | snid 4667 | . . . . . . . 8 ⊢ ∅ ∈ {∅} |
14 | nnex 12270 | . . . . . . . . . . 11 ⊢ ℕ ∈ V | |
15 | 14 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → ℕ ∈ V) |
16 | 15, 1 | ssexd 5330 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V) |
17 | mapdm0 8881 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝐴 ↑m ∅) = {∅}) | |
18 | 16, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑m ∅) = {∅}) |
19 | 13, 18 | eleqtrrid 2846 | . . . . . . 7 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑m ∅)) |
20 | 6 | oveq2i 7442 | . . . . . . 7 ⊢ (𝐴 ↑m (0..^0)) = (𝐴 ↑m ∅) |
21 | 19, 20 | eleqtrrdi 2850 | . . . . . 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 2741 | . . . . 5 ⊢ ((𝜑 ∧ 𝑀 = 0) → 0 = 𝑀) |
25 | 20, 18 | eqtrid 2787 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑m (0..^0)) = {∅}) |
26 | 25 | eleq2d 2825 | . . . . . . . 8 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑m (0..^0)) ↔ 𝑐 ∈ {∅})) |
27 | 26 | biimpa 476 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑐 ∈ {∅}) |
28 | elsni 4648 | . . . . . . 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 4674 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 = 0) → {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀} = {∅}) |
32 | 31 | eqcomd 2741 | . . 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 2945 | . . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 𝑀 ≠ 0) |
36 | 35 | necomd 2994 | . . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → 0 ≠ 𝑀) |
37 | 33, 36 | eqnetrd 3006 | . . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) ≠ 𝑀) |
38 | 37 | neneqd 2943 | . . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑀 = 0) ∧ 𝑐 ∈ (𝐴 ↑m (0..^0))) → ¬ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) |
39 | 38 | ralrimiva 3144 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → ∀𝑐 ∈ (𝐴 ↑m (0..^0)) ¬ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀) |
40 | rabeq0 4394 | . . . . 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 2741 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 = 0) → ∅ = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
43 | 32, 42 | ifeqda 4567 | . 2 ⊢ (𝜑 → if(𝑀 = 0, {∅}, ∅) = {𝑐 ∈ (𝐴 ↑m (0..^0)) ∣ Σ𝑎 ∈ (0..^0)(𝑐‘𝑎) = 𝑀}) |
44 | 5, 43 | eqtr4d 2778 | 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 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 ifcif 4531 {csn 4631 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 0cc0 11153 ℕcn 12264 ℕ0cn0 12524 ℤcz 12611 ..^cfzo 13691 Σcsu 15719 reprcrepr 34602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-repr 34603 |
This theorem is referenced by: breprexp 34627 |
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