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Mirrors > Home > MPE Home > Th. List > 3wlkdlem8 | Structured version Visualization version GIF version |
Description: Lemma 8 for 3wlkd 29211. (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 29207 | . . 3 ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V)) |
7 | s3fv0 14807 | . . . 4 ⊢ (𝐽 ∈ V → (⟨“𝐽𝐾𝐿”⟩‘0) = 𝐽) | |
8 | s3fv1 14808 | . . . 4 ⊢ (𝐾 ∈ V → (⟨“𝐽𝐾𝐿”⟩‘1) = 𝐾) | |
9 | s3fv2 14809 | . . . 4 ⊢ (𝐿 ∈ V → (⟨“𝐽𝐾𝐿”⟩‘2) = 𝐿) | |
10 | 7, 8, 9 | 3anim123i 1151 | . . 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 6863 | . . . 4 ⊢ (𝐹‘0) = (⟨“𝐽𝐾𝐿”⟩‘0) |
13 | 12 | eqeq1i 2736 | . . 3 ⊢ ((𝐹‘0) = 𝐽 ↔ (⟨“𝐽𝐾𝐿”⟩‘0) = 𝐽) |
14 | 2 | fveq1i 6863 | . . . 4 ⊢ (𝐹‘1) = (⟨“𝐽𝐾𝐿”⟩‘1) |
15 | 14 | eqeq1i 2736 | . . 3 ⊢ ((𝐹‘1) = 𝐾 ↔ (⟨“𝐽𝐾𝐿”⟩‘1) = 𝐾) |
16 | 2 | fveq1i 6863 | . . . 4 ⊢ (𝐹‘2) = (⟨“𝐽𝐾𝐿”⟩‘2) |
17 | 16 | eqeq1i 2736 | . . 3 ⊢ ((𝐹‘2) = 𝐿 ↔ (⟨“𝐽𝐾𝐿”⟩‘2) = 𝐿) |
18 | 13, 15, 17 | 3anbi123i 1155 | . 2 ⊢ (((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿) ↔ ((⟨“𝐽𝐾𝐿”⟩‘0) = 𝐽 ∧ (⟨“𝐽𝐾𝐿”⟩‘1) = 𝐾 ∧ (⟨“𝐽𝐾𝐿”⟩‘2) = 𝐿)) |
19 | 11, 18 | sylibr 233 | 1 ⊢ (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 Vcvv 3459 ⊆ wss 3928 {cpr 4608 ‘cfv 6516 0cc0 11075 1c1 11076 2c2 12232 ⟨“cs3 14758 ⟨“cs4 14759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8670 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-card 9899 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-nn 12178 df-2 12240 df-3 12241 df-n0 12438 df-z 12524 df-uz 12788 df-fz 13450 df-fzo 13593 df-hash 14256 df-word 14430 df-concat 14486 df-s1 14511 df-s2 14764 df-s3 14765 df-s4 14766 |
This theorem is referenced by: 3wlkdlem9 29209 |
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