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Mirrors > Home > MPE Home > Th. List > Mathboxes > cgrextendand | Structured version Visualization version GIF version |
Description: Deduction form of cgrextend 35527. (Contributed by Scott Fenton, 14-Oct-2013.) |
Ref | Expression |
---|---|
cgrextendand.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
cgrextendand.2 | ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) |
cgrextendand.3 | ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) |
cgrextendand.4 | ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) |
cgrextendand.5 | ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) |
cgrextendand.6 | ⊢ (𝜑 → 𝐸 ∈ (𝔼‘𝑁)) |
cgrextendand.7 | ⊢ (𝜑 → 𝐹 ∈ (𝔼‘𝑁)) |
cgrextendand.8 | ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn ⟨𝐴, 𝐶⟩) |
cgrextendand.9 | ⊢ ((𝜑 ∧ 𝜓) → 𝐸 Btwn ⟨𝐷, 𝐹⟩) |
cgrextendand.10 | ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩) |
cgrextendand.11 | ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩) |
Ref | Expression |
---|---|
cgrextendand | ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cgrextendand.8 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn ⟨𝐴, 𝐶⟩) | |
2 | cgrextendand.9 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐸 Btwn ⟨𝐷, 𝐹⟩) | |
3 | 1, 2 | jca 511 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐹⟩)) |
4 | cgrextendand.10 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩) | |
5 | cgrextendand.11 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩) | |
6 | 4, 5 | jca 511 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) |
7 | cgrextendand.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
8 | cgrextendand.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) | |
9 | cgrextendand.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) | |
10 | cgrextendand.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) | |
11 | cgrextendand.5 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) | |
12 | cgrextendand.6 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ (𝔼‘𝑁)) | |
13 | cgrextendand.7 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝔼‘𝑁)) | |
14 | cgrextend 35527 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐹⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩)) | |
15 | 7, 8, 9, 10, 11, 12, 13, 14 | syl133anc 1391 | . . 3 ⊢ (𝜑 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐹⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩)) |
16 | 15 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐹⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩)) |
17 | 3, 6, 16 | mp2and 698 | 1 ⊢ ((𝜑 ∧ 𝜓) → ⟨𝐴, 𝐶⟩Cgr⟨𝐷, 𝐹⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2099 ⟨cop 4630 class class class wbr 5142 ‘cfv 6542 ℕcn 12228 𝔼cee 28673 Btwn cbtwn 28674 Cgrccgr 28675 |
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 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
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 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-seq 13985 df-exp 14045 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-sum 15651 df-ee 28676 df-btwn 28677 df-cgr 28678 df-ofs 35502 |
This theorem is referenced by: cgrxfr 35574 btwnconn1lem1 35606 btwnconn1lem2 35607 btwnconn1lem3 35608 btwnconn1lem8 35613 btwnconn1lem10 35615 |
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