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Mirrors > Home > MPE Home > Th. List > Mathboxes > linecgrand | Structured version Visualization version GIF version |
Description: Deduction form of linecgr 35613. (Contributed by Scott Fenton, 14-Oct-2013.) |
Ref | Expression |
---|---|
linecgrand.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
linecgrand.2 | ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) |
linecgrand.3 | ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) |
linecgrand.4 | ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) |
linecgrand.5 | ⊢ (𝜑 → 𝑃 ∈ (𝔼‘𝑁)) |
linecgrand.6 | ⊢ (𝜑 → 𝑄 ∈ (𝔼‘𝑁)) |
linecgrand.7 | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ≠ 𝐵) |
linecgrand.8 | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 Colinear 〈𝐵, 𝐶〉) |
linecgrand.9 | ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝑃〉Cgr〈𝐴, 𝑄〉) |
linecgrand.10 | ⊢ ((𝜑 ∧ 𝜓) → 〈𝐵, 𝑃〉Cgr〈𝐵, 𝑄〉) |
Ref | Expression |
---|---|
linecgrand | ⊢ ((𝜑 ∧ 𝜓) → 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑄〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | linecgrand.7 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ≠ 𝐵) | |
2 | linecgrand.8 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 Colinear 〈𝐵, 𝐶〉) | |
3 | 1, 2 | jca 511 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ≠ 𝐵 ∧ 𝐴 Colinear 〈𝐵, 𝐶〉)) |
4 | linecgrand.9 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 〈𝐴, 𝑃〉Cgr〈𝐴, 𝑄〉) | |
5 | linecgrand.10 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 〈𝐵, 𝑃〉Cgr〈𝐵, 𝑄〉) | |
6 | 4, 5 | jca 511 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (〈𝐴, 𝑃〉Cgr〈𝐴, 𝑄〉 ∧ 〈𝐵, 𝑃〉Cgr〈𝐵, 𝑄〉)) |
7 | linecgrand.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
8 | linecgrand.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) | |
9 | linecgrand.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) | |
10 | linecgrand.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) | |
11 | linecgrand.5 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ (𝔼‘𝑁)) | |
12 | linecgrand.6 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝔼‘𝑁)) | |
13 | linecgr 35613 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁))) → (((𝐴 ≠ 𝐵 ∧ 𝐴 Colinear 〈𝐵, 𝐶〉) ∧ (〈𝐴, 𝑃〉Cgr〈𝐴, 𝑄〉 ∧ 〈𝐵, 𝑃〉Cgr〈𝐵, 𝑄〉)) → 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑄〉)) | |
14 | 7, 8, 9, 10, 11, 12, 13 | syl132anc 1386 | . . 3 ⊢ (𝜑 → (((𝐴 ≠ 𝐵 ∧ 𝐴 Colinear 〈𝐵, 𝐶〉) ∧ (〈𝐴, 𝑃〉Cgr〈𝐴, 𝑄〉 ∧ 〈𝐵, 𝑃〉Cgr〈𝐵, 𝑄〉)) → 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑄〉)) |
15 | 14 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 ≠ 𝐵 ∧ 𝐴 Colinear 〈𝐵, 𝐶〉) ∧ (〈𝐴, 𝑃〉Cgr〈𝐴, 𝑄〉 ∧ 〈𝐵, 𝑃〉Cgr〈𝐵, 𝑄〉)) → 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑄〉)) |
16 | 3, 6, 15 | mp2and 698 | 1 ⊢ ((𝜑 ∧ 𝜓) → 〈𝐶, 𝑃〉Cgr〈𝐶, 𝑄〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ≠ wne 2935 〈cop 4630 class class class wbr 5142 ‘cfv 6542 ℕcn 12234 𝔼cee 28686 Cgrccgr 28688 Colinear ccolin 35569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-ico 13354 df-icc 13355 df-fz 13509 df-fzo 13652 df-seq 13991 df-exp 14051 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-clim 15456 df-sum 15657 df-ee 28689 df-btwn 28690 df-cgr 28691 df-ofs 35515 df-colinear 35571 df-ifs 35572 df-cgr3 35573 df-fs 35574 |
This theorem is referenced by: btwnconn1lem12 35630 |
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