![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > btwnexchand | Structured version Visualization version GIF version |
Description: Deduction form of btwnexch 33102. (Contributed by Scott Fenton, 13-Oct-2013.) |
Ref | Expression |
---|---|
btwnexchand.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
btwnexchand.2 | ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) |
btwnexchand.3 | ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) |
btwnexchand.4 | ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) |
btwnexchand.5 | ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) |
btwnexchand.6 | ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn 〈𝐴, 𝐶〉) |
btwnexchand.7 | ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn 〈𝐴, 𝐷〉) |
Ref | Expression |
---|---|
btwnexchand | ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn 〈𝐴, 𝐷〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwnexchand.6 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn 〈𝐴, 𝐶〉) | |
2 | btwnexchand.7 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn 〈𝐴, 𝐷〉) | |
3 | btwnexchand.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | btwnexchand.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) | |
5 | btwnexchand.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) | |
6 | btwnexchand.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) | |
7 | btwnexchand.5 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) | |
8 | btwnexch 33102 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐵 Btwn 〈𝐴, 𝐷〉)) | |
9 | 3, 4, 5, 6, 7, 8 | syl122anc 1372 | . . 3 ⊢ (𝜑 → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐵 Btwn 〈𝐴, 𝐷〉)) |
10 | 9 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐵 Btwn 〈𝐴, 𝐷〉)) |
11 | 1, 2, 10 | mp2and 695 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn 〈𝐴, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2081 〈cop 4482 class class class wbr 4966 ‘cfv 6230 ℕcn 11491 𝔼cee 26362 Btwn cbtwn 26363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-inf2 8955 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-pre-sup 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-se 5408 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-isom 6239 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-map 8263 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-sup 8757 df-oi 8825 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-2 11553 df-3 11554 df-n0 11751 df-z 11835 df-uz 12099 df-rp 12245 df-ico 12599 df-icc 12600 df-fz 12748 df-fzo 12889 df-seq 13225 df-exp 13285 df-hash 13546 df-cj 14297 df-re 14298 df-im 14299 df-sqrt 14433 df-abs 14434 df-clim 14684 df-sum 14882 df-ee 26365 df-btwn 26366 df-cgr 26367 df-ofs 33060 |
This theorem is referenced by: cgrxfr 33132 btwnconn1lem2 33165 btwnconn1lem12 33175 segletr 33191 segleantisym 33192 outsideoftr 33206 outsideofeq 33207 lineelsb2 33225 |
Copyright terms: Public domain | W3C validator |