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| Mirrors > Home > MPE Home > Th. List > hlcgreulem | Structured version Visualization version GIF version | ||
| Description: Lemma for hlcgreu 28626. (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 726 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐺 ∈ TarskiG) |
| 6 | ishlg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 7 | 6 | ad2antrr 726 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ 𝑃) |
| 8 | ishlg.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 9 | 8 | ad2antrr 726 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐵 ∈ 𝑃) |
| 10 | ishlg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
| 11 | 10 | ad2antrr 726 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐶 ∈ 𝑃) |
| 12 | simplr 769 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑦 ∈ 𝑃) | |
| 13 | hlcgreulem.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
| 14 | 13 | ad2antrr 726 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 ∈ 𝑃) |
| 15 | hlcgreulem.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
| 16 | 15 | ad2antrr 726 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑌 ∈ 𝑃) |
| 17 | simprr 773 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ≠ 𝑦) | |
| 18 | 17 | necomd 2996 | . . 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 726 | . . . . 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 28612 | . . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑋) |
| 24 | 23 | ad2antrr 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑋) |
| 25 | simprl 771 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝐷𝐼𝑦)) | |
| 26 | 1, 3, 19, 21, 14, 12, 5, 7, 24, 25 | btwnhl 28622 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑋𝐼𝑦)) |
| 27 | 1, 2, 3, 5, 14, 7, 12, 26 | tgbtwncom 28496 | . . 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 28612 | . . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑌) |
| 30 | 29 | ad2antrr 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑌) |
| 31 | 1, 3, 19, 21, 16, 12, 5, 7, 30, 25 | btwnhl 28622 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑌𝐼𝑦)) |
| 32 | 1, 2, 3, 5, 16, 7, 12, 31 | tgbtwncom 28496 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑦𝐼𝑌)) |
| 33 | hlcgreulem.3 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶)) | |
| 34 | 33 | ad2antrr 726 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → (𝐴 − 𝑋) = (𝐵 − 𝐶)) |
| 35 | hlcgreulem.4 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶)) | |
| 36 | 35 | ad2antrr 726 | . . 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 28494 | . 2 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 = 𝑌) |
| 38 | 1 | fvexi 6920 | . . . . 5 ⊢ 𝑃 ∈ V |
| 39 | 38 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ V) |
| 40 | hlcgrex.2 | . . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
| 41 | 39, 8, 10, 40 | nehash2 14513 | . . 3 ⊢ (𝜑 → 2 ≤ (♯‘𝑃)) |
| 42 | 1, 2, 3, 4, 20, 6, 41 | tgbtwndiff 28514 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) |
| 43 | 37, 42 | r19.29a 3162 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 hlGchlg 28608 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 df-hlg 28609 |
| This theorem is referenced by: hlcgreu 28626 iscgra1 28818 |
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