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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 12992 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((#‘𝐴) ∈ ℕ0 ∨ (#‘𝐴) = +∞)) | |
2 | df-nel 2783 | . . . . . . . . 9 ⊢ ((#‘𝐴) ∉ ℕ0 ↔ ¬ (#‘𝐴) ∈ ℕ0) | |
3 | 2 | anbi2i 726 | . . . . . . . 8 ⊢ ((((#‘𝐴) = +∞ ∨ (#‘𝐴) ∈ ℕ0) ∧ (#‘𝐴) ∉ ℕ0) ↔ (((#‘𝐴) = +∞ ∨ (#‘𝐴) ∈ ℕ0) ∧ ¬ (#‘𝐴) ∈ ℕ0)) |
4 | pm5.61 745 | . . . . . . . 8 ⊢ ((((#‘𝐴) = +∞ ∨ (#‘𝐴) ∈ ℕ0) ∧ ¬ (#‘𝐴) ∈ ℕ0) ↔ ((#‘𝐴) = +∞ ∧ ¬ (#‘𝐴) ∈ ℕ0)) | |
5 | 3, 4 | sylbb 208 | . . . . . . 7 ⊢ ((((#‘𝐴) = +∞ ∨ (#‘𝐴) ∈ ℕ0) ∧ (#‘𝐴) ∉ ℕ0) → ((#‘𝐴) = +∞ ∧ ¬ (#‘𝐴) ∈ ℕ0)) |
6 | 5 | ex 449 | . . . . . 6 ⊢ (((#‘𝐴) = +∞ ∨ (#‘𝐴) ∈ ℕ0) → ((#‘𝐴) ∉ ℕ0 → ((#‘𝐴) = +∞ ∧ ¬ (#‘𝐴) ∈ ℕ0))) |
7 | 6 | orcoms 403 | . . . . 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 444 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (#‘𝐴) ∉ ℕ0) → ((#‘𝐴) = +∞ ∧ ¬ (#‘𝐴) ∈ ℕ0)) |
10 | 9 | 3adant2 1073 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (#‘𝐴) ∉ ℕ0) → ((#‘𝐴) = +∞ ∧ ¬ (#‘𝐴) ∈ ℕ0)) |
11 | oveq1 6556 | . . . . 5 ⊢ ((#‘𝐴) = +∞ → ((#‘𝐴) +𝑒 (#‘𝐵)) = (+∞ +𝑒 (#‘𝐵))) | |
12 | hashxrcl 13010 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → (#‘𝐵) ∈ ℝ*) | |
13 | hashnemnf 12994 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → (#‘𝐵) ≠ -∞) | |
14 | 12, 13 | jca 553 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → ((#‘𝐵) ∈ ℝ* ∧ (#‘𝐵) ≠ -∞)) |
15 | 14 | 3ad2ant2 1076 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (#‘𝐴) ∉ ℕ0) → ((#‘𝐵) ∈ ℝ* ∧ (#‘𝐵) ≠ -∞)) |
16 | xaddpnf2 11932 | . . . . . 6 ⊢ (((#‘𝐵) ∈ ℝ* ∧ (#‘𝐵) ≠ -∞) → (+∞ +𝑒 (#‘𝐵)) = +∞) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (#‘𝐴) ∉ ℕ0) → (+∞ +𝑒 (#‘𝐵)) = +∞) |
18 | 11, 17 | sylan9eqr 2666 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (#‘𝐴) ∉ ℕ0) ∧ (#‘𝐴) = +∞) → ((#‘𝐴) +𝑒 (#‘𝐵)) = +∞) |
19 | 18 | expcom 450 | . . 3 ⊢ ((#‘𝐴) = +∞ → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (#‘𝐴) ∉ ℕ0) → ((#‘𝐴) +𝑒 (#‘𝐵)) = +∞)) |
20 | 19 | adantr 480 | . 2 ⊢ (((#‘𝐴) = +∞ ∧ ¬ (#‘𝐴) ∈ ℕ0) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (#‘𝐴) ∉ ℕ0) → ((#‘𝐴) +𝑒 (#‘𝐵)) = +∞)) |
21 | 10, 20 | mpcom 37 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (#‘𝐴) ∉ ℕ0) → ((#‘𝐴) +𝑒 (#‘𝐵)) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 ‘cfv 5804 (class class class)co 6549 +∞cpnf 9950 -∞cmnf 9951 ℝ*cxr 9952 ℕ0cn0 11169 +𝑒 cxad 11820 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-xadd 11823 df-hash 12980 |
This theorem is referenced by: hashunx 13036 vdgrf 26425 |
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