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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 12113 | . . . . . . . 8 ⊢ (𝑌 ∈ ℕ0 → (𝑌 + 1) ∈ ℕ0) | |
2 | eleq1a 2829 | . . . . . . . . . . . . 13 ⊢ ((𝑌 + 1) ∈ ℕ0 → ((♯‘𝑉) = (𝑌 + 1) → (♯‘𝑉) ∈ ℕ0)) | |
3 | 2 | adantr 484 | . . . . . . . . . . . 12 ⊢ (((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) → ((♯‘𝑉) = (𝑌 + 1) → (♯‘𝑉) ∈ ℕ0)) |
4 | 3 | imp 410 | . . . . . . . . . . 11 ⊢ ((((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) ∧ (♯‘𝑉) = (𝑌 + 1)) → (♯‘𝑉) ∈ ℕ0) |
5 | hashclb 13908 | . . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (𝑉 ∈ Fin ↔ (♯‘𝑉) ∈ ℕ0)) | |
6 | 5 | ad2antlr 727 | . . . . . . . . . . 11 ⊢ ((((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) ∧ (♯‘𝑉) = (𝑌 + 1)) → (𝑉 ∈ Fin ↔ (♯‘𝑉) ∈ ℕ0)) |
7 | 4, 6 | mpbird 260 | . . . . . . . . . 10 ⊢ ((((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) ∧ (♯‘𝑉) = (𝑌 + 1)) → 𝑉 ∈ Fin) |
8 | 7 | ex 416 | . . . . . . . . 9 ⊢ (((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) → ((♯‘𝑉) = (𝑌 + 1) → 𝑉 ∈ Fin)) |
9 | 8 | ex 416 | . . . . . . . 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 411 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑌 ∈ ℕ0) → ((♯‘𝑉) = (𝑌 + 1) → 𝑉 ∈ Fin)) |
12 | 11 | 3adant2 1133 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((♯‘𝑉) = (𝑌 + 1) → 𝑉 ∈ Fin)) |
13 | 12 | imp 410 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (♯‘𝑉) = (𝑌 + 1)) → 𝑉 ∈ Fin) |
14 | snssi 4711 | . . . . . 6 ⊢ (𝑁 ∈ 𝑉 → {𝑁} ⊆ 𝑉) | |
15 | 14 | 3ad2ant2 1136 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → {𝑁} ⊆ 𝑉) |
16 | 15 | adantr 484 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (♯‘𝑉) = (𝑌 + 1)) → {𝑁} ⊆ 𝑉) |
17 | hashssdif 13962 | . . . 4 ⊢ ((𝑉 ∈ Fin ∧ {𝑁} ⊆ 𝑉) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − (♯‘{𝑁}))) | |
18 | 13, 16, 17 | syl2anc 587 | . . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (♯‘𝑉) = (𝑌 + 1)) → (♯‘(𝑉 ∖ {𝑁})) = ((♯‘𝑉) − (♯‘{𝑁}))) |
19 | oveq1 7209 | . . . 4 ⊢ ((♯‘𝑉) = (𝑌 + 1) → ((♯‘𝑉) − (♯‘{𝑁})) = ((𝑌 + 1) − (♯‘{𝑁}))) | |
20 | hashsng 13919 | . . . . . . 7 ⊢ (𝑁 ∈ 𝑉 → (♯‘{𝑁}) = 1) | |
21 | 20 | oveq2d 7218 | . . . . . 6 ⊢ (𝑁 ∈ 𝑉 → ((𝑌 + 1) − (♯‘{𝑁})) = ((𝑌 + 1) − 1)) |
22 | 21 | 3ad2ant2 1136 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((𝑌 + 1) − (♯‘{𝑁})) = ((𝑌 + 1) − 1)) |
23 | nn0cn 12083 | . . . . . . 7 ⊢ (𝑌 ∈ ℕ0 → 𝑌 ∈ ℂ) | |
24 | 1cnd 10811 | . . . . . . 7 ⊢ (𝑌 ∈ ℕ0 → 1 ∈ ℂ) | |
25 | 23, 24 | pncand 11173 | . . . . . 6 ⊢ (𝑌 ∈ ℕ0 → ((𝑌 + 1) − 1) = 𝑌) |
26 | 25 | 3ad2ant3 1137 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((𝑌 + 1) − 1) = 𝑌) |
27 | 22, 26 | eqtrd 2774 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((𝑌 + 1) − (♯‘{𝑁})) = 𝑌) |
28 | 19, 27 | sylan9eqr 2796 | . . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (♯‘𝑉) = (𝑌 + 1)) → ((♯‘𝑉) − (♯‘{𝑁})) = 𝑌) |
29 | 18, 28 | eqtrd 2774 | . 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (♯‘𝑉) = (𝑌 + 1)) → (♯‘(𝑉 ∖ {𝑁})) = 𝑌) |
30 | 29 | ex 416 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((♯‘𝑉) = (𝑌 + 1) → (♯‘(𝑉 ∖ {𝑁})) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∖ cdif 3854 ⊆ wss 3857 {csn 4531 ‘cfv 6369 (class class class)co 7202 Fincfn 8615 1c1 10713 + caddc 10715 − cmin 11045 ℕ0cn0 12073 ♯chash 13879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-oadd 8195 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-hash 13880 |
This theorem is referenced by: fi1uzind 14046 brfi1indALT 14049 cusgrsize2inds 27513 fsuppind 39941 |
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