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Mirrors > Home > MPE Home > Th. List > hashdifsnp1 | Structured version Visualization version GIF version |
Description: If the size of a set is a nonnegative integer increased by 1, the size of the set with one of its elements removed is this nonnegative integer. (Contributed by Alexander van der Vekens, 7-Jan-2018.) |
Ref | Expression |
---|---|
hashdifsnp1 | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((♯‘𝑉) = (𝑌 + 1) → (♯‘(𝑉 ∖ {𝑁})) = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2nn0 12273 | . . . . . . . 8 ⊢ (𝑌 ∈ ℕ0 → (𝑌 + 1) ∈ ℕ0) | |
2 | eleq1a 2836 | . . . . . . . . . . . . 13 ⊢ ((𝑌 + 1) ∈ ℕ0 → ((♯‘𝑉) = (𝑌 + 1) → (♯‘𝑉) ∈ ℕ0)) | |
3 | 2 | adantr 481 | . . . . . . . . . . . 12 ⊢ (((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) → ((♯‘𝑉) = (𝑌 + 1) → (♯‘𝑉) ∈ ℕ0)) |
4 | 3 | imp 407 | . . . . . . . . . . 11 ⊢ ((((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) ∧ (♯‘𝑉) = (𝑌 + 1)) → (♯‘𝑉) ∈ ℕ0) |
5 | hashclb 14071 | . . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (𝑉 ∈ Fin ↔ (♯‘𝑉) ∈ ℕ0)) | |
6 | 5 | ad2antlr 724 | . . . . . . . . . . 11 ⊢ ((((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) ∧ (♯‘𝑉) = (𝑌 + 1)) → (𝑉 ∈ Fin ↔ (♯‘𝑉) ∈ ℕ0)) |
7 | 4, 6 | mpbird 256 | . . . . . . . . . 10 ⊢ ((((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) ∧ (♯‘𝑉) = (𝑌 + 1)) → 𝑉 ∈ Fin) |
8 | 7 | ex 413 | . . . . . . . . 9 ⊢ (((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) → ((♯‘𝑉) = (𝑌 + 1) → 𝑉 ∈ Fin)) |
9 | 8 | ex 413 | . . . . . . . 8 ⊢ ((𝑌 + 1) ∈ ℕ0 → (𝑉 ∈ 𝑊 → ((♯‘𝑉) = (𝑌 + 1) → 𝑉 ∈ Fin))) |
10 | 1, 9 | syl 17 | . . . . . . 7 ⊢ (𝑌 ∈ ℕ0 → (𝑉 ∈ 𝑊 → ((♯‘𝑉) = (𝑌 + 1) → 𝑉 ∈ Fin))) |
11 | 10 | impcom 408 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑌 ∈ ℕ0) → ((♯‘𝑉) = (𝑌 + 1) → 𝑉 ∈ Fin)) |
12 | 11 | 3adant2 1130 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((♯‘𝑉) = (𝑌 + 1) → 𝑉 ∈ Fin)) |
13 | 12 | imp 407 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (♯‘𝑉) = (𝑌 + 1)) → 𝑉 ∈ Fin) |
14 | snssi 4747 | . . . . . 6 ⊢ (𝑁 ∈ 𝑉 → {𝑁} ⊆ 𝑉) | |
15 | 14 | 3ad2ant2 1133 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → {𝑁} ⊆ 𝑉) |
16 | 15 | adantr 481 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (♯‘𝑉) = (𝑌 + 1)) → {𝑁} ⊆ 𝑉) |
17 | hashssdif 14125 | . . . 4 ⊢ ((𝑉 ∈ Fin ∧ {𝑁} ⊆ 𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − (♯‘{𝑁}))) | |
18 | 13, 16, 17 | syl2anc 584 | . . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (♯‘𝑉) = (𝑌 + 1)) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − (♯‘{𝑁}))) |
19 | oveq1 7278 | . . . 4 ⊢ ((♯‘𝑉) = (𝑌 + 1) → ((♯‘𝑉) − (♯‘{𝑁})) = ((𝑌 + 1) − (♯‘{𝑁}))) | |
20 | hashsng 14082 | . . . . . . 7 ⊢ (𝑁 ∈ 𝑉 → (♯‘{𝑁}) = 1) | |
21 | 20 | oveq2d 7287 | . . . . . 6 ⊢ (𝑁 ∈ 𝑉 → ((𝑌 + 1) − (♯‘{𝑁})) = ((𝑌 + 1) − 1)) |
22 | 21 | 3ad2ant2 1133 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((𝑌 + 1) − (♯‘{𝑁})) = ((𝑌 + 1) − 1)) |
23 | nn0cn 12243 | . . . . . . 7 ⊢ (𝑌 ∈ ℕ0 → 𝑌 ∈ ℂ) | |
24 | 1cnd 10971 | . . . . . . 7 ⊢ (𝑌 ∈ ℕ0 → 1 ∈ ℂ) | |
25 | 23, 24 | pncand 11333 | . . . . . 6 ⊢ (𝑌 ∈ ℕ0 → ((𝑌 + 1) − 1) = 𝑌) |
26 | 25 | 3ad2ant3 1134 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((𝑌 + 1) − 1) = 𝑌) |
27 | 22, 26 | eqtrd 2780 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((𝑌 + 1) − (♯‘{𝑁})) = 𝑌) |
28 | 19, 27 | sylan9eqr 2802 | . . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (♯‘𝑉) = (𝑌 + 1)) → ((♯‘𝑉) − (♯‘{𝑁})) = 𝑌) |
29 | 18, 28 | eqtrd 2780 | . 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (♯‘𝑉) = (𝑌 + 1)) → (♯‘(𝑉 ∖ {𝑁})) = 𝑌) |
30 | 29 | ex 413 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((♯‘𝑉) = (𝑌 + 1) → (♯‘(𝑉 ∖ {𝑁})) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ∖ cdif 3889 ⊆ wss 3892 {csn 4567 ‘cfv 6432 (class class class)co 7271 Fincfn 8716 1c1 10873 + caddc 10875 − cmin 11205 ℕ0cn0 12233 ♯chash 14042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-oadd 8292 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-dju 9660 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-hash 14043 |
This theorem is referenced by: fi1uzind 14209 brfi1indALT 14212 cusgrsize2inds 27818 fsuppind 40276 |
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