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Mirrors > Home > MPE Home > Th. List > 2wlkdlem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for 2wlkd 27402. (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)) |
2wlkd.n | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) |
Ref | Expression |
---|---|
2wlkdlem5 | ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2wlkd.n | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) | |
2 | 2wlkd.p | . . . . 5 ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 | |
3 | 2wlkd.f | . . . . 5 ⊢ 𝐹 = 〈“𝐽𝐾”〉 | |
4 | 2wlkd.s | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | |
5 | 2, 3, 4 | 2wlkdlem3 27393 | . . . 4 ⊢ (𝜑 → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) |
6 | simp1 1129 | . . . . . . 7 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘0) = 𝐴) | |
7 | simp2 1130 | . . . . . . 7 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘1) = 𝐵) | |
8 | 6, 7 | neeq12d 3045 | . . . . . 6 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘0) ≠ (𝑃‘1) ↔ 𝐴 ≠ 𝐵)) |
9 | simp3 1131 | . . . . . . 7 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) = 𝐶) | |
10 | 7, 9 | neeq12d 3045 | . . . . . 6 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘1) ≠ (𝑃‘2) ↔ 𝐵 ≠ 𝐶)) |
11 | 8, 10 | anbi12d 630 | . . . . 5 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2)) ↔ (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))) |
12 | 11 | bicomd 224 | . . . 4 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ↔ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2)))) |
13 | 5, 12 | syl 17 | . . 3 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ↔ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2)))) |
14 | 1, 13 | mpbid 233 | . 2 ⊢ (𝜑 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2))) |
15 | 2, 3 | 2wlkdlem2 27392 | . . . 4 ⊢ (0..^(♯‘𝐹)) = {0, 1} |
16 | 15 | raleqi 3373 | . . 3 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ∀𝑘 ∈ {0, 1} (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) |
17 | c0ex 10481 | . . . 4 ⊢ 0 ∈ V | |
18 | 1ex 10483 | . . . 4 ⊢ 1 ∈ V | |
19 | fveq2 6538 | . . . . 5 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0)) | |
20 | fv0p1e1 11608 | . . . . 5 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1)) | |
21 | 19, 20 | neeq12d 3045 | . . . 4 ⊢ (𝑘 = 0 → ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ (𝑃‘0) ≠ (𝑃‘1))) |
22 | fveq2 6538 | . . . . 5 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1)) | |
23 | oveq1 7023 | . . . . . . 7 ⊢ (𝑘 = 1 → (𝑘 + 1) = (1 + 1)) | |
24 | 1p1e2 11610 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
25 | 23, 24 | syl6eq 2847 | . . . . . 6 ⊢ (𝑘 = 1 → (𝑘 + 1) = 2) |
26 | 25 | fveq2d 6542 | . . . . 5 ⊢ (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2)) |
27 | 22, 26 | neeq12d 3045 | . . . 4 ⊢ (𝑘 = 1 → ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ (𝑃‘1) ≠ (𝑃‘2))) |
28 | 17, 18, 21, 27 | ralpr 4543 | . . 3 ⊢ (∀𝑘 ∈ {0, 1} (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2))) |
29 | 16, 28 | bitri 276 | . 2 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2))) |
30 | 14, 29 | sylibr 235 | 1 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 {cpr 4474 ‘cfv 6225 (class class class)co 7016 0cc0 10383 1c1 10384 + caddc 10386 2c2 11540 ..^cfzo 12883 ♯chash 13540 〈“cs2 14039 〈“cs3 14040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-n0 11746 df-z 11830 df-uz 12094 df-fz 12743 df-fzo 12884 df-hash 13541 df-word 13708 df-concat 13769 df-s1 13794 df-s2 14046 df-s3 14047 |
This theorem is referenced by: 2wlkd 27402 |
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