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Mirrors > Home > MPE Home > Th. List > 3wlkdlem8 | Structured version Visualization version GIF version |
Description: Lemma 8 for 3wlkd 27943. (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 27939 | . . 3 ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V)) |
7 | s3fv0 14247 | . . . 4 ⊢ (𝐽 ∈ V → (〈“𝐽𝐾𝐿”〉‘0) = 𝐽) | |
8 | s3fv1 14248 | . . . 4 ⊢ (𝐾 ∈ V → (〈“𝐽𝐾𝐿”〉‘1) = 𝐾) | |
9 | s3fv2 14249 | . . . 4 ⊢ (𝐿 ∈ V → (〈“𝐽𝐾𝐿”〉‘2) = 𝐿) | |
10 | 7, 8, 9 | 3anim123i 1147 | . . 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 6666 | . . . 4 ⊢ (𝐹‘0) = (〈“𝐽𝐾𝐿”〉‘0) |
13 | 12 | eqeq1i 2826 | . . 3 ⊢ ((𝐹‘0) = 𝐽 ↔ (〈“𝐽𝐾𝐿”〉‘0) = 𝐽) |
14 | 2 | fveq1i 6666 | . . . 4 ⊢ (𝐹‘1) = (〈“𝐽𝐾𝐿”〉‘1) |
15 | 14 | eqeq1i 2826 | . . 3 ⊢ ((𝐹‘1) = 𝐾 ↔ (〈“𝐽𝐾𝐿”〉‘1) = 𝐾) |
16 | 2 | fveq1i 6666 | . . . 4 ⊢ (𝐹‘2) = (〈“𝐽𝐾𝐿”〉‘2) |
17 | 16 | eqeq1i 2826 | . . 3 ⊢ ((𝐹‘2) = 𝐿 ↔ (〈“𝐽𝐾𝐿”〉‘2) = 𝐿) |
18 | 13, 15, 17 | 3anbi123i 1151 | . 2 ⊢ (((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿) ↔ ((〈“𝐽𝐾𝐿”〉‘0) = 𝐽 ∧ (〈“𝐽𝐾𝐿”〉‘1) = 𝐾 ∧ (〈“𝐽𝐾𝐿”〉‘2) = 𝐿)) |
19 | 11, 18 | sylibr 236 | 1 ⊢ (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3495 ⊆ wss 3936 {cpr 4563 ‘cfv 6350 0cc0 10531 1c1 10532 2c2 11686 〈“cs3 14198 〈“cs4 14199 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-s3 14205 df-s4 14206 |
This theorem is referenced by: 3wlkdlem9 27941 |
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