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Mirrors > Home > MPE Home > Th. List > Mathboxes > btwncomand | Structured version Visualization version GIF version |
Description: Deduction form of btwncom 33972. (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 33972 | . . . 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 484 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 Btwn 〈𝐵, 𝐶〉 ↔ 𝐴 Btwn 〈𝐶, 𝐵〉)) |
9 | 1, 8 | mpbid 235 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 Btwn 〈𝐶, 𝐵〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2114 〈cop 4532 class class class wbr 5040 ‘cfv 6350 ℕcn 11729 𝔼cee 26847 Btwn cbtwn 26848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-inf2 9190 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-sup 8992 df-oi 9060 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-n0 11990 df-z 12076 df-uz 12338 df-rp 12486 df-ico 12840 df-icc 12841 df-fz 12995 df-fzo 13138 df-seq 13474 df-exp 13535 df-hash 13796 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-clim 14948 df-sum 15149 df-ee 26850 df-btwn 26851 df-cgr 26852 |
This theorem is referenced by: btwnouttr 33982 cgrxfr 34013 btwnconn1lem1 34045 btwnconn1lem2 34046 btwnconn1lem3 34047 btwnconn1lem8 34052 btwnconn1lem10 34054 btwnconn1lem11 34055 btwnconn3 34061 segcon2 34063 segleantisym 34073 colinbtwnle 34076 broutsideof2 34080 btwnoutside 34083 broutsideof3 34084 outsidele 34090 lineunray 34105 lineelsb2 34106 |
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