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Mirrors > Home > MPE Home > Th. List > 2wlkdlem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for 2wlkd 29966. (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 29957 | . . . 4 ⊢ (𝜑 → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) |
6 | simp1 1135 | . . . . . . 7 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘0) = 𝐴) | |
7 | simp2 1136 | . . . . . . 7 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘1) = 𝐵) | |
8 | 6, 7 | neeq12d 3000 | . . . . . 6 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘0) ≠ (𝑃‘1) ↔ 𝐴 ≠ 𝐵)) |
9 | simp3 1137 | . . . . . . 7 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → (𝑃‘2) = 𝐶) | |
10 | 7, 9 | neeq12d 3000 | . . . . . 6 ⊢ (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶) → ((𝑃‘1) ≠ (𝑃‘2) ↔ 𝐵 ≠ 𝐶)) |
11 | 8, 10 | anbi12d 632 | . . . . 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 223 | . . . 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 232 | . 2 ⊢ (𝜑 → ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2))) |
15 | 2, 3 | 2wlkdlem2 29956 | . . . 4 ⊢ (0..^(♯‘𝐹)) = {0, 1} |
16 | 15 | raleqi 3322 | . . 3 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ∀𝑘 ∈ {0, 1} (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) |
17 | c0ex 11253 | . . . 4 ⊢ 0 ∈ V | |
18 | 1ex 11255 | . . . 4 ⊢ 1 ∈ V | |
19 | fveq2 6907 | . . . . 5 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0)) | |
20 | fv0p1e1 12387 | . . . . 5 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1)) | |
21 | 19, 20 | neeq12d 3000 | . . . 4 ⊢ (𝑘 = 0 → ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ (𝑃‘0) ≠ (𝑃‘1))) |
22 | fveq2 6907 | . . . . 5 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1)) | |
23 | oveq1 7438 | . . . . . . 7 ⊢ (𝑘 = 1 → (𝑘 + 1) = (1 + 1)) | |
24 | 1p1e2 12389 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
25 | 23, 24 | eqtrdi 2791 | . . . . . 6 ⊢ (𝑘 = 1 → (𝑘 + 1) = 2) |
26 | 25 | fveq2d 6911 | . . . . 5 ⊢ (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2)) |
27 | 22, 26 | neeq12d 3000 | . . . 4 ⊢ (𝑘 = 1 → ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ (𝑃‘1) ≠ (𝑃‘2))) |
28 | 17, 18, 21, 27 | ralpr 4705 | . . 3 ⊢ (∀𝑘 ∈ {0, 1} (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2))) |
29 | 16, 28 | bitri 275 | . 2 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘1) ≠ (𝑃‘2))) |
30 | 14, 29 | sylibr 234 | 1 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 {cpr 4633 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 2c2 12319 ..^cfzo 13691 ♯chash 14366 〈“cs2 14877 〈“cs3 14878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-s2 14884 df-s3 14885 |
This theorem is referenced by: 2wlkd 29966 |
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