| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cntnevol | Structured version Visualization version GIF version | ||
| Description: Counting and Lebesgue measures are different. (Contributed by Thierry Arnoux, 27-Jan-2017.) |
| Ref | Expression |
|---|---|
| cntnevol | ⊢ (♯ ↾ 𝒫 𝑂) ≠ vol |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-1ne0 11224 | . . . . 5 ⊢ 1 ≠ 0 | |
| 2 | 1 | a1i 11 | . . . 4 ⊢ (1 ∈ 𝑂 → 1 ≠ 0) |
| 3 | snelpwi 5448 | . . . . . 6 ⊢ (1 ∈ 𝑂 → {1} ∈ 𝒫 𝑂) | |
| 4 | fvres 6925 | . . . . . 6 ⊢ ({1} ∈ 𝒫 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = (♯‘{1})) | |
| 5 | 3, 4 | syl 17 | . . . . 5 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = (♯‘{1})) |
| 6 | 1re 11261 | . . . . . 6 ⊢ 1 ∈ ℝ | |
| 7 | hashsng 14408 | . . . . . 6 ⊢ (1 ∈ ℝ → (♯‘{1}) = 1) | |
| 8 | 6, 7 | ax-mp 5 | . . . . 5 ⊢ (♯‘{1}) = 1 |
| 9 | 5, 8 | eqtrdi 2793 | . . . 4 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = 1) |
| 10 | snssi 4808 | . . . . . . 7 ⊢ (1 ∈ ℝ → {1} ⊆ ℝ) | |
| 11 | ovolsn 25530 | . . . . . . 7 ⊢ (1 ∈ ℝ → (vol*‘{1}) = 0) | |
| 12 | nulmbl 25570 | . . . . . . 7 ⊢ (({1} ⊆ ℝ ∧ (vol*‘{1}) = 0) → {1} ∈ dom vol) | |
| 13 | 10, 11, 12 | syl2anc 584 | . . . . . 6 ⊢ (1 ∈ ℝ → {1} ∈ dom vol) |
| 14 | mblvol 25565 | . . . . . . 7 ⊢ ({1} ∈ dom vol → (vol‘{1}) = (vol*‘{1})) | |
| 15 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ (vol*‘{1}) = 0 |
| 16 | 14, 15 | eqtrdi 2793 | . . . . . 6 ⊢ ({1} ∈ dom vol → (vol‘{1}) = 0) |
| 17 | 6, 13, 16 | mp2b 10 | . . . . 5 ⊢ (vol‘{1}) = 0 |
| 18 | 17 | a1i 11 | . . . 4 ⊢ (1 ∈ 𝑂 → (vol‘{1}) = 0) |
| 19 | 2, 9, 18 | 3netr4d 3018 | . . 3 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) ≠ (vol‘{1})) |
| 20 | fveq1 6905 | . . . 4 ⊢ ((♯ ↾ 𝒫 𝑂) = vol → ((♯ ↾ 𝒫 𝑂)‘{1}) = (vol‘{1})) | |
| 21 | 20 | necon3i 2973 | . . 3 ⊢ (((♯ ↾ 𝒫 𝑂)‘{1}) ≠ (vol‘{1}) → (♯ ↾ 𝒫 𝑂) ≠ vol) |
| 22 | 19, 21 | syl 17 | . 2 ⊢ (1 ∈ 𝑂 → (♯ ↾ 𝒫 𝑂) ≠ vol) |
| 23 | 6, 13 | ax-mp 5 | . . . . . . 7 ⊢ {1} ∈ dom vol |
| 24 | 23 | biantrur 530 | . . . . . 6 ⊢ (¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ ({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂))) |
| 25 | snex 5436 | . . . . . . . . 9 ⊢ {1} ∈ V | |
| 26 | 25 | elpw 4604 | . . . . . . . 8 ⊢ ({1} ∈ 𝒫 𝑂 ↔ {1} ⊆ 𝑂) |
| 27 | dmhashres 14380 | . . . . . . . . 9 ⊢ dom (♯ ↾ 𝒫 𝑂) = 𝒫 𝑂 | |
| 28 | 27 | eleq2i 2833 | . . . . . . . 8 ⊢ ({1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ {1} ∈ 𝒫 𝑂) |
| 29 | 1ex 11257 | . . . . . . . . 9 ⊢ 1 ∈ V | |
| 30 | 29 | snss 4785 | . . . . . . . 8 ⊢ (1 ∈ 𝑂 ↔ {1} ⊆ 𝑂) |
| 31 | 26, 28, 30 | 3bitr4i 303 | . . . . . . 7 ⊢ ({1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ 1 ∈ 𝑂) |
| 32 | 31 | notbii 320 | . . . . . 6 ⊢ (¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ ¬ 1 ∈ 𝑂) |
| 33 | 24, 32 | bitr3i 277 | . . . . 5 ⊢ (({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂)) ↔ ¬ 1 ∈ 𝑂) |
| 34 | nelne1 3039 | . . . . 5 ⊢ (({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂)) → dom vol ≠ dom (♯ ↾ 𝒫 𝑂)) | |
| 35 | 33, 34 | sylbir 235 | . . . 4 ⊢ (¬ 1 ∈ 𝑂 → dom vol ≠ dom (♯ ↾ 𝒫 𝑂)) |
| 36 | 35 | necomd 2996 | . . 3 ⊢ (¬ 1 ∈ 𝑂 → dom (♯ ↾ 𝒫 𝑂) ≠ dom vol) |
| 37 | dmeq 5914 | . . . 4 ⊢ ((♯ ↾ 𝒫 𝑂) = vol → dom (♯ ↾ 𝒫 𝑂) = dom vol) | |
| 38 | 37 | necon3i 2973 | . . 3 ⊢ (dom (♯ ↾ 𝒫 𝑂) ≠ dom vol → (♯ ↾ 𝒫 𝑂) ≠ vol) |
| 39 | 36, 38 | syl 17 | . 2 ⊢ (¬ 1 ∈ 𝑂 → (♯ ↾ 𝒫 𝑂) ≠ vol) |
| 40 | 22, 39 | pm2.61i 182 | 1 ⊢ (♯ ↾ 𝒫 𝑂) ≠ vol |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 𝒫 cpw 4600 {csn 4626 dom cdm 5685 ↾ cres 5687 ‘cfv 6561 ℝcr 11154 0cc0 11155 1c1 11156 ♯chash 14369 vol*covol 25497 volcvol 25498 |
| 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-of 7697 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-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 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-xnn0 12600 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xadd 13155 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 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-xmet 21357 df-met 21358 df-ovol 25499 df-vol 25500 |
| This theorem is referenced by: (None) |
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