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Mirrors > Home > MPE Home > Th. List > 3wlkdlem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for 3wlkd 29690. (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 6891 | . . . . 5 ⊢ (𝑃‘0) = (⟨“𝐴𝐵𝐶𝐷”⟩‘0) |
4 | s4fv0 14850 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘0) = 𝐴) | |
5 | 3, 4 | eqtrid 2782 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
6 | 2 | fveq1i 6891 | . . . . 5 ⊢ (𝑃‘1) = (⟨“𝐴𝐵𝐶𝐷”⟩‘1) |
7 | s4fv1 14851 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘1) = 𝐵) | |
8 | 6, 7 | eqtrid 2782 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑃‘1) = 𝐵) |
9 | 5, 8 | anim12i 611 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵)) |
10 | 2 | fveq1i 6891 | . . . . 5 ⊢ (𝑃‘2) = (⟨“𝐴𝐵𝐶𝐷”⟩‘2) |
11 | s4fv2 14852 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘2) = 𝐶) | |
12 | 10, 11 | eqtrid 2782 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝑃‘2) = 𝐶) |
13 | 2 | fveq1i 6891 | . . . . 5 ⊢ (𝑃‘3) = (⟨“𝐴𝐵𝐶𝐷”⟩‘3) |
14 | s4fv3 14853 | . . . . 5 ⊢ (𝐷 ∈ 𝑉 → (⟨“𝐴𝐵𝐶𝐷”⟩‘3) = 𝐷) | |
15 | 13, 14 | eqtrid 2782 | . . . 4 ⊢ (𝐷 ∈ 𝑉 → (𝑃‘3) = 𝐷) |
16 | 12, 15 | anim12i 611 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐷)) |
17 | 9, 16 | anim12i 611 | . 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 394 = wceq 1539 ∈ wcel 2104 ‘cfv 6542 0cc0 11112 1c1 11113 2c2 12271 3c3 12272 ⟨“cs3 14797 ⟨“cs4 14798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-s4 14805 |
This theorem is referenced by: 3wlkdlem4 29682 3wlkdlem5 29683 3pthdlem1 29684 3wlkdlem6 29685 3wlkdlem10 29689 3wlkond 29691 |
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