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Mirrors > Home > MPE Home > Th. List > 3wlkdlem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for 3wlkd 29390. (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 6882 | . . . . 5 ⊢ (𝑃‘0) = (〈“𝐴𝐵𝐶𝐷”〉‘0) |
4 | s4fv0 14833 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘0) = 𝐴) | |
5 | 3, 4 | eqtrid 2785 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑃‘0) = 𝐴) |
6 | 2 | fveq1i 6882 | . . . . 5 ⊢ (𝑃‘1) = (〈“𝐴𝐵𝐶𝐷”〉‘1) |
7 | s4fv1 14834 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘1) = 𝐵) | |
8 | 6, 7 | eqtrid 2785 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑃‘1) = 𝐵) |
9 | 5, 8 | anim12i 614 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵)) |
10 | 2 | fveq1i 6882 | . . . . 5 ⊢ (𝑃‘2) = (〈“𝐴𝐵𝐶𝐷”〉‘2) |
11 | s4fv2 14835 | . . . . 5 ⊢ (𝐶 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘2) = 𝐶) | |
12 | 10, 11 | eqtrid 2785 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → (𝑃‘2) = 𝐶) |
13 | 2 | fveq1i 6882 | . . . . 5 ⊢ (𝑃‘3) = (〈“𝐴𝐵𝐶𝐷”〉‘3) |
14 | s4fv3 14836 | . . . . 5 ⊢ (𝐷 ∈ 𝑉 → (〈“𝐴𝐵𝐶𝐷”〉‘3) = 𝐷) | |
15 | 13, 14 | eqtrid 2785 | . . . 4 ⊢ (𝐷 ∈ 𝑉 → (𝑃‘3) = 𝐷) |
16 | 12, 15 | anim12i 614 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐷)) |
17 | 9, 16 | anim12i 614 | . 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 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6535 0cc0 11097 1c1 11098 2c2 12254 3c3 12255 〈“cs3 14780 〈“cs4 14781 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-n0 12460 df-z 12546 df-uz 12810 df-fz 13472 df-fzo 13615 df-hash 14278 df-word 14452 df-concat 14508 df-s1 14533 df-s2 14786 df-s3 14787 df-s4 14788 |
This theorem is referenced by: 3wlkdlem4 29382 3wlkdlem5 29383 3pthdlem1 29384 3wlkdlem6 29385 3wlkdlem10 29389 3wlkond 29391 |
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