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Mirrors > Home > MPE Home > Th. List > Mathboxes > btwncomand | Structured version Visualization version GIF version |
Description: Deduction form of btwncom 34645. (Contributed by Scott Fenton, 14-Oct-2013.) |
Ref | Expression |
---|---|
btwncomand.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
btwncomand.2 | ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) |
btwncomand.3 | ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) |
btwncomand.4 | ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) |
btwncomand.5 | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 Btwn ⟨𝐵, 𝐶⟩) |
Ref | Expression |
---|---|
btwncomand | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 Btwn ⟨𝐶, 𝐵⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwncomand.5 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 Btwn ⟨𝐵, 𝐶⟩) | |
2 | btwncomand.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
3 | btwncomand.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) | |
4 | btwncomand.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) | |
5 | btwncomand.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) | |
6 | btwncom 34645 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐶, 𝐵⟩)) | |
7 | 2, 3, 4, 5, 6 | syl13anc 1373 | . . 3 ⊢ (𝜑 → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐶, 𝐵⟩)) |
8 | 7 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐶, 𝐵⟩)) |
9 | 1, 8 | mpbid 231 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 Btwn ⟨𝐶, 𝐵⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ⟨cop 4593 class class class wbr 5106 ‘cfv 6497 ℕcn 12158 𝔼cee 27879 Btwn cbtwn 27880 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-sum 15577 df-ee 27882 df-btwn 27883 df-cgr 27884 |
This theorem is referenced by: btwnouttr 34655 cgrxfr 34686 btwnconn1lem1 34718 btwnconn1lem2 34719 btwnconn1lem3 34720 btwnconn1lem8 34725 btwnconn1lem10 34727 btwnconn1lem11 34728 btwnconn3 34734 segcon2 34736 segleantisym 34746 colinbtwnle 34749 broutsideof2 34753 btwnoutside 34756 broutsideof3 34757 outsidele 34763 lineunray 34778 lineelsb2 34779 |
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