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Mirrors > Home > MPE Home > Th. List > hashxrcl | Structured version Visualization version GIF version |
Description: Extended real closure of the ♯ function. (Contributed by Mario Carneiro, 22-Apr-2015.) |
Ref | Expression |
---|---|
hashxrcl | ⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ssre 11496 | . . . 4 ⊢ ℕ0 ⊆ ℝ | |
2 | ressxr 10283 | . . . 4 ⊢ ℝ ⊆ ℝ* | |
3 | 1, 2 | sstri 3761 | . . 3 ⊢ ℕ0 ⊆ ℝ* |
4 | pnfxr 10292 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | snssi 4474 | . . . 4 ⊢ (+∞ ∈ ℝ* → {+∞} ⊆ ℝ*) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ {+∞} ⊆ ℝ* |
7 | 3, 6 | unssi 3939 | . 2 ⊢ (ℕ0 ∪ {+∞}) ⊆ ℝ* |
8 | elex 3364 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
9 | hashf 13322 | . . . 4 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
10 | 9 | ffvelrni 6499 | . . 3 ⊢ (𝐴 ∈ V → (♯‘𝐴) ∈ (ℕ0 ∪ {+∞})) |
11 | 8, 10 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ∈ (ℕ0 ∪ {+∞})) |
12 | 7, 11 | sseldi 3750 | 1 ⊢ (𝐴 ∈ 𝑉 → (♯‘𝐴) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 Vcvv 3351 ∪ cun 3721 ⊆ wss 3723 {csn 4316 ‘cfv 6029 ℝcr 10135 +∞cpnf 10271 ℝ*cxr 10273 ℕ0cn0 11492 ♯chash 13314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-card 8963 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-nn 11221 df-n0 11493 df-xnn0 11564 df-z 11578 df-uz 11887 df-hash 13315 |
This theorem is referenced by: nfile 13345 hashdom 13363 hashinfxadd 13369 hashunx 13370 hashgt0 13372 hashle00 13383 hashgt0elex 13384 hashss 13392 hashgt12el 13405 hashgt12el2 13406 ramtlecl 15904 0ram 15924 isnzr2hash 19472 0ringnnzr 19477 ewlkle 26729 upgrewlkle2 26730 esumcst 30458 esumpinfval 30468 idomodle 38293 |
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