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Mirrors > Home > MPE Home > Th. List > hashinfxadd | Structured version Visualization version GIF version |
Description: The extended real addition of the size of an infinite set with the size of an arbitrary set yields plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.) |
Ref | Expression |
---|---|
hashinfxadd | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashnn0pnf 14249 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞)) | |
2 | df-nel 3051 | . . . . . . . . 9 ⊢ ((♯‘𝐴) ∉ ℕ0 ↔ ¬ (♯‘𝐴) ∈ ℕ0) | |
3 | 2 | anbi2i 624 | . . . . . . . 8 ⊢ ((((♯‘𝐴) = +∞ ∨ (♯‘𝐴) ∈ ℕ0) ∧ (♯‘𝐴) ∉ ℕ0) ↔ (((♯‘𝐴) = +∞ ∨ (♯‘𝐴) ∈ ℕ0) ∧ ¬ (♯‘𝐴) ∈ ℕ0)) |
4 | pm5.61 1000 | . . . . . . . 8 ⊢ ((((♯‘𝐴) = +∞ ∨ (♯‘𝐴) ∈ ℕ0) ∧ ¬ (♯‘𝐴) ∈ ℕ0) ↔ ((♯‘𝐴) = +∞ ∧ ¬ (♯‘𝐴) ∈ ℕ0)) | |
5 | 3, 4 | sylbb 218 | . . . . . . 7 ⊢ ((((♯‘𝐴) = +∞ ∨ (♯‘𝐴) ∈ ℕ0) ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) = +∞ ∧ ¬ (♯‘𝐴) ∈ ℕ0)) |
6 | 5 | ex 414 | . . . . . 6 ⊢ (((♯‘𝐴) = +∞ ∨ (♯‘𝐴) ∈ ℕ0) → ((♯‘𝐴) ∉ ℕ0 → ((♯‘𝐴) = +∞ ∧ ¬ (♯‘𝐴) ∈ ℕ0))) |
7 | 6 | orcoms 871 | . . . . 5 ⊢ (((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞) → ((♯‘𝐴) ∉ ℕ0 → ((♯‘𝐴) = +∞ ∧ ¬ (♯‘𝐴) ∈ ℕ0))) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) ∉ ℕ0 → ((♯‘𝐴) = +∞ ∧ ¬ (♯‘𝐴) ∈ ℕ0))) |
9 | 8 | imp 408 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) = +∞ ∧ ¬ (♯‘𝐴) ∈ ℕ0)) |
10 | 9 | 3adant2 1132 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) = +∞ ∧ ¬ (♯‘𝐴) ∈ ℕ0)) |
11 | oveq1 7369 | . . . . 5 ⊢ ((♯‘𝐴) = +∞ → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = (+∞ +𝑒 (♯‘𝐵))) | |
12 | hashxrcl 14264 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → (♯‘𝐵) ∈ ℝ*) | |
13 | hashnemnf 14251 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → (♯‘𝐵) ≠ -∞) | |
14 | 12, 13 | jca 513 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → ((♯‘𝐵) ∈ ℝ* ∧ (♯‘𝐵) ≠ -∞)) |
15 | 14 | 3ad2ant2 1135 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐵) ∈ ℝ* ∧ (♯‘𝐵) ≠ -∞)) |
16 | xaddpnf2 13153 | . . . . . 6 ⊢ (((♯‘𝐵) ∈ ℝ* ∧ (♯‘𝐵) ≠ -∞) → (+∞ +𝑒 (♯‘𝐵)) = +∞) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → (+∞ +𝑒 (♯‘𝐵)) = +∞) |
18 | 11, 17 | sylan9eqr 2799 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) ∧ (♯‘𝐴) = +∞) → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = +∞) |
19 | 18 | expcom 415 | . . 3 ⊢ ((♯‘𝐴) = +∞ → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = +∞)) |
20 | 19 | adantr 482 | . 2 ⊢ (((♯‘𝐴) = +∞ ∧ ¬ (♯‘𝐴) ∈ ℕ0) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = +∞)) |
21 | 10, 20 | mpcom 38 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∉ wnel 3050 ‘cfv 6501 (class class class)co 7362 +∞cpnf 11193 -∞cmnf 11194 ℝ*cxr 11195 ℕ0cn0 12420 +𝑒 cxad 13038 ♯chash 14237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-xadd 13041 df-hash 14238 |
This theorem is referenced by: hashunx 14293 |
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