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Mirrors > Home > MPE Home > Th. List > brfi1indlem | Structured version Visualization version GIF version |
Description: Lemma for brfi1ind 13136: The size of a set is the size of this set with one element removed, increased by 1. (Contributed by Alexander van der Vekens, 7-Jan-2018.) |
Ref | Expression |
---|---|
brfi1indlem | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((#‘𝑉) = (𝑌 + 1) → (#‘(𝑉 ∖ {𝑁})) = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2nn0 11210 | . . . . . . . 8 ⊢ (𝑌 ∈ ℕ0 → (𝑌 + 1) ∈ ℕ0) | |
2 | eleq1a 2683 | . . . . . . . . . . . . 13 ⊢ ((𝑌 + 1) ∈ ℕ0 → ((#‘𝑉) = (𝑌 + 1) → (#‘𝑉) ∈ ℕ0)) | |
3 | 2 | adantr 480 | . . . . . . . . . . . 12 ⊢ (((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) → ((#‘𝑉) = (𝑌 + 1) → (#‘𝑉) ∈ ℕ0)) |
4 | 3 | imp 444 | . . . . . . . . . . 11 ⊢ ((((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) ∧ (#‘𝑉) = (𝑌 + 1)) → (#‘𝑉) ∈ ℕ0) |
5 | hashclb 13011 | . . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝑊 → (𝑉 ∈ Fin ↔ (#‘𝑉) ∈ ℕ0)) | |
6 | 5 | ad2antlr 759 | . . . . . . . . . . 11 ⊢ ((((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) ∧ (#‘𝑉) = (𝑌 + 1)) → (𝑉 ∈ Fin ↔ (#‘𝑉) ∈ ℕ0)) |
7 | 4, 6 | mpbird 246 | . . . . . . . . . 10 ⊢ ((((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) ∧ (#‘𝑉) = (𝑌 + 1)) → 𝑉 ∈ Fin) |
8 | 7 | ex 449 | . . . . . . . . 9 ⊢ (((𝑌 + 1) ∈ ℕ0 ∧ 𝑉 ∈ 𝑊) → ((#‘𝑉) = (𝑌 + 1) → 𝑉 ∈ Fin)) |
9 | 8 | ex 449 | . . . . . . . 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 445 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑌 ∈ ℕ0) → ((#‘𝑉) = (𝑌 + 1) → 𝑉 ∈ Fin)) |
12 | 11 | 3adant2 1073 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((#‘𝑉) = (𝑌 + 1) → 𝑉 ∈ Fin)) |
13 | 12 | imp 444 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (#‘𝑉) = (𝑌 + 1)) → 𝑉 ∈ Fin) |
14 | snssi 4280 | . . . . . 6 ⊢ (𝑁 ∈ 𝑉 → {𝑁} ⊆ 𝑉) | |
15 | 14 | 3ad2ant2 1076 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → {𝑁} ⊆ 𝑉) |
16 | 15 | adantr 480 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (#‘𝑉) = (𝑌 + 1)) → {𝑁} ⊆ 𝑉) |
17 | hashssdif 13061 | . . . 4 ⊢ ((𝑉 ∈ Fin ∧ {𝑁} ⊆ 𝑉) → (#‘(𝑉 ∖ {𝑁})) = ((#‘𝑉) − (#‘{𝑁}))) | |
18 | 13, 16, 17 | syl2anc 691 | . . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (#‘𝑉) = (𝑌 + 1)) → (#‘(𝑉 ∖ {𝑁})) = ((#‘𝑉) − (#‘{𝑁}))) |
19 | oveq1 6556 | . . . 4 ⊢ ((#‘𝑉) = (𝑌 + 1) → ((#‘𝑉) − (#‘{𝑁})) = ((𝑌 + 1) − (#‘{𝑁}))) | |
20 | hashsng 13020 | . . . . . . 7 ⊢ (𝑁 ∈ 𝑉 → (#‘{𝑁}) = 1) | |
21 | 20 | oveq2d 6565 | . . . . . 6 ⊢ (𝑁 ∈ 𝑉 → ((𝑌 + 1) − (#‘{𝑁})) = ((𝑌 + 1) − 1)) |
22 | 21 | 3ad2ant2 1076 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((𝑌 + 1) − (#‘{𝑁})) = ((𝑌 + 1) − 1)) |
23 | nn0cn 11179 | . . . . . . 7 ⊢ (𝑌 ∈ ℕ0 → 𝑌 ∈ ℂ) | |
24 | 1cnd 9935 | . . . . . . 7 ⊢ (𝑌 ∈ ℕ0 → 1 ∈ ℂ) | |
25 | 23, 24 | pncand 10272 | . . . . . 6 ⊢ (𝑌 ∈ ℕ0 → ((𝑌 + 1) − 1) = 𝑌) |
26 | 25 | 3ad2ant3 1077 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((𝑌 + 1) − 1) = 𝑌) |
27 | 22, 26 | eqtrd 2644 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((𝑌 + 1) − (#‘{𝑁})) = 𝑌) |
28 | 19, 27 | sylan9eqr 2666 | . . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (#‘𝑉) = (𝑌 + 1)) → ((#‘𝑉) − (#‘{𝑁})) = 𝑌) |
29 | 18, 28 | eqtrd 2644 | . 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) ∧ (#‘𝑉) = (𝑌 + 1)) → (#‘(𝑉 ∖ {𝑁})) = 𝑌) |
30 | 29 | ex 449 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0) → ((#‘𝑉) = (𝑌 + 1) → (#‘(𝑉 ∖ {𝑁})) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 1c1 9816 + caddc 9818 − cmin 10145 ℕ0cn0 11169 #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-rep 4699 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-rmo 2904 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-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 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-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 |
This theorem is referenced by: fi1uzind 13134 brfi1indALT 13137 fi1uzindOLD 13140 brfi1indALTOLD 13143 cusgrasize2inds 26005 cusgrsize2inds 40669 |
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