Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 3wlkdlem8 | Structured version Visualization version GIF version |
Description: Lemma 8 for 3wlkd 28520. (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 28516 | . . 3 ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V)) |
7 | s3fv0 14592 | . . . 4 ⊢ (𝐽 ∈ V → (〈“𝐽𝐾𝐿”〉‘0) = 𝐽) | |
8 | s3fv1 14593 | . . . 4 ⊢ (𝐾 ∈ V → (〈“𝐽𝐾𝐿”〉‘1) = 𝐾) | |
9 | s3fv2 14594 | . . . 4 ⊢ (𝐿 ∈ V → (〈“𝐽𝐾𝐿”〉‘2) = 𝐿) | |
10 | 7, 8, 9 | 3anim123i 1150 | . . 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 6768 | . . . 4 ⊢ (𝐹‘0) = (〈“𝐽𝐾𝐿”〉‘0) |
13 | 12 | eqeq1i 2743 | . . 3 ⊢ ((𝐹‘0) = 𝐽 ↔ (〈“𝐽𝐾𝐿”〉‘0) = 𝐽) |
14 | 2 | fveq1i 6768 | . . . 4 ⊢ (𝐹‘1) = (〈“𝐽𝐾𝐿”〉‘1) |
15 | 14 | eqeq1i 2743 | . . 3 ⊢ ((𝐹‘1) = 𝐾 ↔ (〈“𝐽𝐾𝐿”〉‘1) = 𝐾) |
16 | 2 | fveq1i 6768 | . . . 4 ⊢ (𝐹‘2) = (〈“𝐽𝐾𝐿”〉‘2) |
17 | 16 | eqeq1i 2743 | . . 3 ⊢ ((𝐹‘2) = 𝐿 ↔ (〈“𝐽𝐾𝐿”〉‘2) = 𝐿) |
18 | 13, 15, 17 | 3anbi123i 1154 | . 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 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3430 ⊆ wss 3887 {cpr 4564 ‘cfv 6427 0cc0 10859 1c1 10860 2c2 12016 〈“cs3 14543 〈“cs4 14544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7704 df-1st 7821 df-2nd 7822 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-nn 11962 df-2 12024 df-3 12025 df-n0 12222 df-z 12308 df-uz 12571 df-fz 13228 df-fzo 13371 df-hash 14033 df-word 14206 df-concat 14262 df-s1 14289 df-s2 14549 df-s3 14550 df-s4 14551 |
This theorem is referenced by: 3wlkdlem9 28518 |
Copyright terms: Public domain | W3C validator |