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Mirrors > Home > MPE Home > Th. List > 3wlkdlem8 | Structured version Visualization version GIF version |
Description: Lemma 8 for 3wlkd 27542. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
Ref | Expression |
---|---|
3wlkd.p | ⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 |
3wlkd.f | ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 |
3wlkd.s | ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) |
3wlkd.n | ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) |
3wlkd.e | ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) |
Ref | Expression |
---|---|
3wlkdlem8 | ⊢ (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3wlkd.p | . . . 4 ⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 | |
2 | 3wlkd.f | . . . 4 ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 | |
3 | 3wlkd.s | . . . 4 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) | |
4 | 3wlkd.n | . . . 4 ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) | |
5 | 3wlkd.e | . . . 4 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) | |
6 | 1, 2, 3, 4, 5 | 3wlkdlem7 27538 | . . 3 ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V)) |
7 | s3fv0 14019 | . . . 4 ⊢ (𝐽 ∈ V → (〈“𝐽𝐾𝐿”〉‘0) = 𝐽) | |
8 | s3fv1 14020 | . . . 4 ⊢ (𝐾 ∈ V → (〈“𝐽𝐾𝐿”〉‘1) = 𝐾) | |
9 | s3fv2 14021 | . . . 4 ⊢ (𝐿 ∈ V → (〈“𝐽𝐾𝐿”〉‘2) = 𝐿) | |
10 | 7, 8, 9 | 3anim123i 1194 | . . 3 ⊢ ((𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V) → ((〈“𝐽𝐾𝐿”〉‘0) = 𝐽 ∧ (〈“𝐽𝐾𝐿”〉‘1) = 𝐾 ∧ (〈“𝐽𝐾𝐿”〉‘2) = 𝐿)) |
11 | 6, 10 | syl 17 | . 2 ⊢ (𝜑 → ((〈“𝐽𝐾𝐿”〉‘0) = 𝐽 ∧ (〈“𝐽𝐾𝐿”〉‘1) = 𝐾 ∧ (〈“𝐽𝐾𝐿”〉‘2) = 𝐿)) |
12 | 2 | fveq1i 6438 | . . . 4 ⊢ (𝐹‘0) = (〈“𝐽𝐾𝐿”〉‘0) |
13 | 12 | eqeq1i 2830 | . . 3 ⊢ ((𝐹‘0) = 𝐽 ↔ (〈“𝐽𝐾𝐿”〉‘0) = 𝐽) |
14 | 2 | fveq1i 6438 | . . . 4 ⊢ (𝐹‘1) = (〈“𝐽𝐾𝐿”〉‘1) |
15 | 14 | eqeq1i 2830 | . . 3 ⊢ ((𝐹‘1) = 𝐾 ↔ (〈“𝐽𝐾𝐿”〉‘1) = 𝐾) |
16 | 2 | fveq1i 6438 | . . . 4 ⊢ (𝐹‘2) = (〈“𝐽𝐾𝐿”〉‘2) |
17 | 16 | eqeq1i 2830 | . . 3 ⊢ ((𝐹‘2) = 𝐿 ↔ (〈“𝐽𝐾𝐿”〉‘2) = 𝐿) |
18 | 13, 15, 17 | 3anbi123i 1198 | . 2 ⊢ (((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿) ↔ ((〈“𝐽𝐾𝐿”〉‘0) = 𝐽 ∧ (〈“𝐽𝐾𝐿”〉‘1) = 𝐾 ∧ (〈“𝐽𝐾𝐿”〉‘2) = 𝐿)) |
19 | 11, 18 | sylibr 226 | 1 ⊢ (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 Vcvv 3414 ⊆ wss 3798 {cpr 4401 ‘cfv 6127 0cc0 10259 1c1 10260 2c2 11413 〈“cs3 13970 〈“cs4 13971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-hash 13418 df-word 13582 df-concat 13638 df-s1 13663 df-s2 13976 df-s3 13977 df-s4 13978 |
This theorem is referenced by: 3wlkdlem9 27540 |
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