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Mirrors > Home > MPE Home > Th. List > Mathboxes > btwncomand | Structured version Visualization version GIF version |
Description: Deduction form of btwncom 35291. (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 35291 | . . . 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 1371 | . . 3 ⊢ (𝜑 → (𝐴 Btwn 〈𝐵, 𝐶〉 ↔ 𝐴 Btwn 〈𝐶, 𝐵〉)) |
8 | 7 | adantr 480 | . 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 395 ∈ wcel 2105 〈cop 4634 class class class wbr 5148 ‘cfv 6543 ℕcn 12217 𝔼cee 28414 Btwn cbtwn 28415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-rp 12980 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-ee 28417 df-btwn 28418 df-cgr 28419 |
This theorem is referenced by: btwnouttr 35301 cgrxfr 35332 btwnconn1lem1 35364 btwnconn1lem2 35365 btwnconn1lem3 35366 btwnconn1lem8 35371 btwnconn1lem10 35373 btwnconn1lem11 35374 btwnconn3 35380 segcon2 35382 segleantisym 35392 colinbtwnle 35395 broutsideof2 35399 btwnoutside 35402 broutsideof3 35403 outsidele 35409 lineunray 35424 lineelsb2 35425 |
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