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Mirrors > Home > MPE Home > Th. List > Mathboxes > linecgrand | Structured version Visualization version GIF version |
Description: Deduction form of linecgr 35734. (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 510 | . 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 510 | . 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 35734 | . . . 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 1385 | . . 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 479 | . 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 697 | 1 ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐶, 𝑃⟩Cgr⟨𝐶, 𝑄⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ≠ wne 2930 ⟨cop 4630 class class class wbr 5143 ‘cfv 6543 ℕcn 12242 𝔼cee 28743 Cgrccgr 28745 Colinear ccolin 35690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-sum 15665 df-ee 28746 df-btwn 28747 df-cgr 28748 df-ofs 35636 df-colinear 35692 df-ifs 35693 df-cgr3 35694 df-fs 35695 |
This theorem is referenced by: btwnconn1lem12 35751 |
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