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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 14328 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) ∈ ℕ0 ∨ (♯‘𝐴) = +∞)) | |
2 | df-nel 3037 | . . . . . . . . 9 ⊢ ((♯‘𝐴) ∉ ℕ0 ↔ ¬ (♯‘𝐴) ∈ ℕ0) | |
3 | 2 | anbi2i 621 | . . . . . . . 8 ⊢ ((((♯‘𝐴) = +∞ ∨ (♯‘𝐴) ∈ ℕ0) ∧ (♯‘𝐴) ∉ ℕ0) ↔ (((♯‘𝐴) = +∞ ∨ (♯‘𝐴) ∈ ℕ0) ∧ ¬ (♯‘𝐴) ∈ ℕ0)) |
4 | pm5.61 998 | . . . . . . . 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 411 | . . . . . 6 ⊢ (((♯‘𝐴) = +∞ ∨ (♯‘𝐴) ∈ ℕ0) → ((♯‘𝐴) ∉ ℕ0 → ((♯‘𝐴) = +∞ ∧ ¬ (♯‘𝐴) ∈ ℕ0))) |
7 | 6 | orcoms 870 | . . . . 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 405 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) = +∞ ∧ ¬ (♯‘𝐴) ∈ ℕ0)) |
10 | 9 | 3adant2 1128 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) = +∞ ∧ ¬ (♯‘𝐴) ∈ ℕ0)) |
11 | oveq1 7420 | . . . . 5 ⊢ ((♯‘𝐴) = +∞ → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = (+∞ +𝑒 (♯‘𝐵))) | |
12 | hashxrcl 14343 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → (♯‘𝐵) ∈ ℝ*) | |
13 | hashnemnf 14330 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → (♯‘𝐵) ≠ -∞) | |
14 | 12, 13 | jca 510 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → ((♯‘𝐵) ∈ ℝ* ∧ (♯‘𝐵) ≠ -∞)) |
15 | 14 | 3ad2ant2 1131 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐵) ∈ ℝ* ∧ (♯‘𝐵) ≠ -∞)) |
16 | xaddpnf2 13233 | . . . . . 6 ⊢ (((♯‘𝐵) ∈ ℝ* ∧ (♯‘𝐵) ≠ -∞) → (+∞ +𝑒 (♯‘𝐵)) = +∞) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → (+∞ +𝑒 (♯‘𝐵)) = +∞) |
18 | 11, 17 | sylan9eqr 2787 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) ∧ (♯‘𝐴) = +∞) → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = +∞) |
19 | 18 | expcom 412 | . . 3 ⊢ ((♯‘𝐴) = +∞ → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (♯‘𝐴) ∉ ℕ0) → ((♯‘𝐴) +𝑒 (♯‘𝐵)) = +∞)) |
20 | 19 | adantr 479 | . 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 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∉ wnel 3036 ‘cfv 6543 (class class class)co 7413 +∞cpnf 11270 -∞cmnf 11271 ℝ*cxr 11272 ℕ0cn0 12497 +𝑒 cxad 13117 ♯chash 14316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-xnn0 12570 df-z 12584 df-uz 12848 df-xadd 13120 df-hash 14317 |
This theorem is referenced by: hashunx 14372 |
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