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Mirrors > Home > MPE Home > Th. List > hlcgreulem | Structured version Visualization version GIF version |
Description: Lemma for hlcgreu 26406. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
Ref | Expression |
---|---|
ishlg.p | ⊢ 𝑃 = (Base‘𝐺) |
ishlg.i | ⊢ 𝐼 = (Itv‘𝐺) |
ishlg.k | ⊢ 𝐾 = (hlG‘𝐺) |
ishlg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
ishlg.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
ishlg.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
hlln.1 | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
hltr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
hlcgrex.m | ⊢ − = (dist‘𝐺) |
hlcgrex.1 | ⊢ (𝜑 → 𝐷 ≠ 𝐴) |
hlcgrex.2 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
hlcgreulem.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
hlcgreulem.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
hlcgreulem.1 | ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝐷) |
hlcgreulem.2 | ⊢ (𝜑 → 𝑌(𝐾‘𝐴)𝐷) |
hlcgreulem.3 | ⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶)) |
hlcgreulem.4 | ⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶)) |
Ref | Expression |
---|---|
hlcgreulem | ⊢ (𝜑 → 𝑋 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishlg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | hlcgrex.m | . . 3 ⊢ − = (dist‘𝐺) | |
3 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | hlln.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐺 ∈ TarskiG) |
6 | ishlg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ 𝑃) |
8 | ishlg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | 8 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐵 ∈ 𝑃) |
10 | ishlg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
11 | 10 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐶 ∈ 𝑃) |
12 | simplr 767 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑦 ∈ 𝑃) | |
13 | hlcgreulem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
14 | 13 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 ∈ 𝑃) |
15 | hlcgreulem.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
16 | 15 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑌 ∈ 𝑃) |
17 | simprr 771 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ≠ 𝑦) | |
18 | 17 | necomd 3073 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑦 ≠ 𝐴) |
19 | ishlg.k | . . . . 5 ⊢ 𝐾 = (hlG‘𝐺) | |
20 | hltr.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
21 | 20 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷 ∈ 𝑃) |
22 | hlcgreulem.1 | . . . . . . 7 ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝐷) | |
23 | 1, 3, 19, 13, 20, 6, 4, 22 | hlcomd 26392 | . . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑋) |
24 | 23 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑋) |
25 | simprl 769 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝐷𝐼𝑦)) | |
26 | 1, 3, 19, 21, 14, 12, 5, 7, 24, 25 | btwnhl 26402 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑋𝐼𝑦)) |
27 | 1, 2, 3, 5, 14, 7, 12, 26 | tgbtwncom 26276 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑦𝐼𝑋)) |
28 | hlcgreulem.2 | . . . . . . 7 ⊢ (𝜑 → 𝑌(𝐾‘𝐴)𝐷) | |
29 | 1, 3, 19, 15, 20, 6, 4, 28 | hlcomd 26392 | . . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑌) |
30 | 29 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑌) |
31 | 1, 3, 19, 21, 16, 12, 5, 7, 30, 25 | btwnhl 26402 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑌𝐼𝑦)) |
32 | 1, 2, 3, 5, 16, 7, 12, 31 | tgbtwncom 26276 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑦𝐼𝑌)) |
33 | hlcgreulem.3 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶)) | |
34 | 33 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → (𝐴 − 𝑋) = (𝐵 − 𝐶)) |
35 | hlcgreulem.4 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶)) | |
36 | 35 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → (𝐴 − 𝑌) = (𝐵 − 𝐶)) |
37 | 1, 2, 3, 5, 7, 9, 11, 12, 14, 16, 18, 27, 32, 34, 36 | tgsegconeq 26274 | . 2 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 = 𝑌) |
38 | 1 | fvexi 6686 | . . . . 5 ⊢ 𝑃 ∈ V |
39 | 38 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ V) |
40 | hlcgrex.2 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
41 | 39, 8, 10, 40 | nehash2 13835 | . . 3 ⊢ (𝜑 → 2 ≤ (♯‘𝑃)) |
42 | 1, 2, 3, 4, 20, 6, 41 | tgbtwndiff 26294 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) |
43 | 37, 42 | r19.29a 3291 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 distcds 16576 TarskiGcstrkg 26218 Itvcitv 26224 hlGchlg 26388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 df-trkgc 26236 df-trkgb 26237 df-trkgcb 26238 df-trkg 26241 df-hlg 26389 |
This theorem is referenced by: hlcgreu 26406 iscgra1 26598 |
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