![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > seglerflx | Structured version Visualization version GIF version |
Description: Segment comparison is reflexive. Theorem 5.7 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.) |
Ref | Expression |
---|---|
seglerflx | ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 〈𝐴, 𝐵〉 Seg≤ 〈𝐴, 𝐵〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1139 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁)) | |
2 | btwntriv2 34915 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐵 Btwn 〈𝐴, 𝐵〉) | |
3 | cgrrflx 34890 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉) | |
4 | breq1 5147 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 Btwn 〈𝐴, 𝐵〉 ↔ 𝐵 Btwn 〈𝐴, 𝐵〉)) | |
5 | opeq2 4870 | . . . . . 6 ⊢ (𝑦 = 𝐵 → 〈𝐴, 𝑦〉 = 〈𝐴, 𝐵〉) | |
6 | 5 | breq2d 5156 | . . . . 5 ⊢ (𝑦 = 𝐵 → (〈𝐴, 𝐵〉Cgr〈𝐴, 𝑦〉 ↔ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉)) |
7 | 4, 6 | anbi12d 632 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝑦 Btwn 〈𝐴, 𝐵〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑦〉) ↔ (𝐵 Btwn 〈𝐴, 𝐵〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉))) |
8 | 7 | rspcev 3611 | . . 3 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ (𝐵 Btwn 〈𝐴, 𝐵〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝐵〉)) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn 〈𝐴, 𝐵〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑦〉)) |
9 | 1, 2, 3, 8 | syl12anc 836 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn 〈𝐴, 𝐵〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑦〉)) |
10 | simp1 1137 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ) | |
11 | simp2 1138 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝐴 ∈ (𝔼‘𝑁)) | |
12 | brsegle 35011 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (〈𝐴, 𝐵〉 Seg≤ 〈𝐴, 𝐵〉 ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn 〈𝐴, 𝐵〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑦〉))) | |
13 | 10, 11, 1, 11, 1, 12 | syl122anc 1380 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (〈𝐴, 𝐵〉 Seg≤ 〈𝐴, 𝐵〉 ↔ ∃𝑦 ∈ (𝔼‘𝑁)(𝑦 Btwn 〈𝐴, 𝐵〉 ∧ 〈𝐴, 𝐵〉Cgr〈𝐴, 𝑦〉))) |
14 | 9, 13 | mpbird 257 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 〈𝐴, 𝐵〉 Seg≤ 〈𝐴, 𝐵〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 〈cop 4630 class class class wbr 5144 ‘cfv 6535 ℕcn 12199 𝔼cee 28113 Btwn cbtwn 28114 Cgrccgr 28115 Seg≤ csegle 35009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-inf2 9623 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 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 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9424 df-oi 9492 df-card 9921 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-n0 12460 df-z 12546 df-uz 12810 df-rp 12962 df-ico 13317 df-icc 13318 df-fz 13472 df-fzo 13615 df-seq 13954 df-exp 14015 df-hash 14278 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-clim 15419 df-sum 15620 df-ee 28116 df-btwn 28117 df-cgr 28118 df-segle 35010 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |