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Mirrors > Home > MPE Home > Th. List > 2wlkdlem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for 2wlkd 29699. (Contributed by AV, 14-Feb-2021.) |
Ref | Expression |
---|---|
2wlkd.p | ⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩ |
2wlkd.f | ⊢ 𝐹 = ⟨“𝐽𝐾”⟩ |
2wlkd.s | ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
Ref | Expression |
---|---|
2wlkdlem4 | ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2wlkd.s | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
2 | 2wlkd.p | . . . . 5 ⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩ | |
3 | 2wlkd.f | . . . . 5 ⊢ 𝐹 = ⟨“𝐽𝐾”⟩ | |
4 | 2, 3, 1 | 2wlkdlem3 29690 | . . . 4 ⊢ (𝜑 → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) |
5 | simp1 1133 | . . . . . . 7 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘0) = 𝐴) | |
6 | 5 | eleq1d 2812 | . . . . . 6 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘0) ∈ 𝑉 ↔ 𝐴 ∈ 𝑉)) |
7 | simp2 1134 | . . . . . . 7 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘1) = 𝐵) | |
8 | 7 | eleq1d 2812 | . . . . . 6 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘1) ∈ 𝑉 ↔ 𝐵 ∈ 𝑉)) |
9 | simp3 1135 | . . . . . . 7 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) = 𝐶) | |
10 | 9 | eleq1d 2812 | . . . . . 6 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘2) ∈ 𝑉 ↔ 𝐶 ∈ 𝑉)) |
11 | 6, 8, 10 | 3anbi123d 1432 | . . . . 5 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉) ↔ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))) |
12 | 11 | bicomd 222 | . . . 4 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ↔ ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉))) |
13 | 4, 12 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ↔ ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉))) |
14 | 1, 13 | mpbid 231 | . 2 ⊢ (𝜑 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)) |
15 | 3 | fveq2i 6888 | . . . . . . 7 ⊢ (♯‘𝐹) = (♯‘⟨“𝐽𝐾”⟩) |
16 | s2len 14846 | . . . . . . 7 ⊢ (♯‘⟨“𝐽𝐾”⟩) = 2 | |
17 | 15, 16 | eqtri 2754 | . . . . . 6 ⊢ (♯‘𝐹) = 2 |
18 | 17 | oveq2i 7416 | . . . . 5 ⊢ (0...(♯‘𝐹)) = (0...2) |
19 | fz0tp 13608 | . . . . 5 ⊢ (0...2) = {0, 1, 2} | |
20 | 18, 19 | eqtri 2754 | . . . 4 ⊢ (0...(♯‘𝐹)) = {0, 1, 2} |
21 | 20 | raleqi 3317 | . . 3 ⊢ (∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉 ↔ ∀𝑘 ∈ {0, 1, 2} (𝑃‘𝑘) ∈ 𝑉) |
22 | c0ex 11212 | . . . 4 ⊢ 0 ∈ V | |
23 | 1ex 11214 | . . . 4 ⊢ 1 ∈ V | |
24 | 2ex 12293 | . . . 4 ⊢ 2 ∈ V | |
25 | fveq2 6885 | . . . . 5 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0)) | |
26 | 25 | eleq1d 2812 | . . . 4 ⊢ (𝑘 = 0 → ((𝑃‘𝑘) ∈ 𝑉 ↔ (𝑃‘0) ∈ 𝑉)) |
27 | fveq2 6885 | . . . . 5 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1)) | |
28 | 27 | eleq1d 2812 | . . . 4 ⊢ (𝑘 = 1 → ((𝑃‘𝑘) ∈ 𝑉 ↔ (𝑃‘1) ∈ 𝑉)) |
29 | fveq2 6885 | . . . . 5 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2)) | |
30 | 29 | eleq1d 2812 | . . . 4 ⊢ (𝑘 = 2 → ((𝑃‘𝑘) ∈ 𝑉 ↔ (𝑃‘2) ∈ 𝑉)) |
31 | 22, 23, 24, 26, 28, 30 | raltp 4704 | . . 3 ⊢ (∀𝑘 ∈ {0, 1, 2} (𝑃‘𝑘) ∈ 𝑉 ↔ ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)) |
32 | 21, 31 | bitri 275 | . 2 ⊢ (∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉 ↔ ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘1) ∈ 𝑉 ∧ (𝑃‘2) ∈ 𝑉)) |
33 | 14, 32 | sylibr 233 | 1 ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 {ctp 4627 ‘cfv 6537 (class class class)co 7405 0cc0 11112 1c1 11113 2c2 12271 ...cfz 13490 ♯chash 14295 ⟨“cs2 14798 ⟨“cs3 14799 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-concat 14527 df-s1 14552 df-s2 14805 df-s3 14806 |
This theorem is referenced by: 2wlkd 29699 |
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