![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 3wlkdlem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for 3wlkd 27955. (Contributed by Alexander van der Vekens, 10-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))) |
Ref | Expression |
---|---|
3wlkdlem3 | ⊢ (𝜑 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3wlkd.s | . 2 ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) | |
2 | 3wlkd.p | . . . . . 6 ⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 | |
3 | 2 | fveq1i 6646 | . . . . 5 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶𝐷”〉‘0) |
4 | s4fv0 14248 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘0) = 𝐴) | |
5 | 3, 4 | syl5eq 2845 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
6 | 2 | fveq1i 6646 | . . . . 5 ⊢ (𝑃‘1) = (〈“𝐴𝐵𝐶𝐷”〉‘1) |
7 | s4fv1 14249 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘1) = 𝐵) | |
8 | 6, 7 | syl5eq 2845 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑃‘1) = 𝐵) |
9 | 5, 8 | anim12i 615 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵)) |
10 | 2 | fveq1i 6646 | . . . . 5 ⊢ (𝑃‘2) = (〈“𝐴𝐵𝐶𝐷”〉‘2) |
11 | s4fv2 14250 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘2) = 𝐶) | |
12 | 10, 11 | syl5eq 2845 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝑃‘2) = 𝐶) |
13 | 2 | fveq1i 6646 | . . . . 5 ⊢ (𝑃‘3) = (〈“𝐴𝐵𝐶𝐷”〉‘3) |
14 | s4fv3 14251 | . . . . 5 ⊢ (𝐷 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘3) = 𝐷) | |
15 | 13, 14 | syl5eq 2845 | . . . 4 ⊢ (𝐷 ∈ 𝑉 → (𝑃‘3) = 𝐷) |
16 | 12, 15 | anim12i 615 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐷)) |
17 | 9, 16 | anim12i 615 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐷))) |
18 | 1, 17 | syl 17 | 1 ⊢ (𝜑 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 0cc0 10526 1c1 10527 2c2 11680 3c3 11681 〈“cs3 14195 〈“cs4 14196 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-s2 14201 df-s3 14202 df-s4 14203 |
This theorem is referenced by: 3wlkdlem4 27947 3wlkdlem5 27948 3pthdlem1 27949 3wlkdlem6 27950 3wlkdlem10 27954 3wlkond 27956 |
Copyright terms: Public domain | W3C validator |