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Theorem symquadprlng 29445

Description: Symmetrical quadrilaterals are parallelograms. Theorem 12.18 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)

**Hypotheses**
Ref	Expression
symquadprlng.p	⊢ 𝑃 = (Base‘𝐺)
symquadprlng.d	⊢ − = (dist‘𝐺)
symquadprlng.l	⊢ 𝐿 = (LineG‘𝐺)
symquadprlng.r	⊢ ∥ = (parlnG‘𝐺)
symquadprlng.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
symquadprlng.1	⊢ (𝜑 → 𝐺 ∈ TarskiG_E)
symquadprlng.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
symquadprlng.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
symquadprlng.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
symquadprlng.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
symquadprlng.2	⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))
symquadprlng.3	⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))
symquadprlng.4	⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
symquadprlng.5	⊢ (𝜑 → 𝑌 ≠ 𝑊)
symquadprlng.6	⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑍))
symquadprlng.7	⊢ (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊))

**Assertion**
Ref	Expression
symquadprlng	⊢ (𝜑 → ((𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊) ∧ (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋)))

**Proof of Theorem symquadprlng**
Step	Hyp	Ref	Expression
1		symquadprlng.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		symquadprlng.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
3		eqid 2760	. . 3 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		symquadprlng.r	. . 3 ⊢ ∥ = (parlnG‘𝐺)
5		eqid 2760	. . 3 ⊢ (midG‘𝐺) = (midG‘𝐺)
6		symquadprlng.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		symquadprlng.1	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG_E)
8		symquadprlng.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
9		symquadprlng.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
10		symquadprlng.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
11		eqid 2760	. . . . . 6 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
12		symquadprlng.4	. . . . . 6 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
13	1, 2, 11, 6, 9, 10, 8, 12	ncolrot2 29031	. . . . 5 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
14	13	orsild 1019	. . . 4 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
15	10, 14	eldifd 4050	. . 3 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
16		symquadprlng.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
17		symquadprlng.d	. . . . . 6 ⊢ − = (dist‘𝐺)
18		eqid 2760	. . . . . 6 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
19		eqid 2760	. . . . . 6 ⊢ ((pInvG‘𝐺)‘𝑇) = ((pInvG‘𝐺)‘𝑇)
20		symquadprlng.5	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑊)
21	1, 11, 2, 6, 9, 16, 20	tgelrnln 29103	. . . . . . 7 ⊢ (𝜑 → (𝑌𝐿𝑊) ∈ ran 𝐿)
22		symquadprlng.7	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊))
23	1, 2, 11, 6, 21, 22	tglnpt 29017	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
24		symquadprlng.2	. . . . . 6 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))
25		symquadprlng.3	. . . . . 6 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))
26		symquadprlng.6	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑍))
27	26	orcd 887	. . . . . 6 ⊢ (𝜑 → (𝑇 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))
28	22	orcd 887	. . . . . 6 ⊢ (𝜑 → (𝑇 ∈ (𝑌𝐿𝑊) ∨ 𝑌 = 𝑊))
29	1, 17, 11, 2, 18, 6, 19, 8, 9, 10, 16, 23, 12, 20, 24, 25, 27, 28	symquadlem 29166	. . . . 5 ⊢ (𝜑 → 𝑋 = (((pInvG‘𝐺)‘𝑇)‘𝑍))
30	1, 2, 11, 6, 9, 10, 8, 12	ncoltgdim2 29033	. . . . . 6 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
31	1, 17, 11, 6, 30, 10, 8, 18, 23	ismidb 29288	. . . . 5 ⊢ (𝜑 → (𝑋 = (((pInvG‘𝐺)‘𝑇)‘𝑍) ↔ (𝑍(midG‘𝐺)𝑋) = 𝑇))
32	29, 31	mpbid 235	. . . 4 ⊢ (𝜑 → (𝑍(midG‘𝐺)𝑋) = 𝑇)
33	1, 17, 11, 6, 30, 8, 10	midcom 29292	. . . 4 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) = (𝑍(midG‘𝐺)𝑋))
34	1, 17, 2, 6, 8, 9, 10, 16, 24, 25, 12, 20, 26, 22	symquadprlnglem 29170	. . . . . 6 ⊢ (𝜑 → ¬ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
35	1, 2, 11, 6, 8, 10, 26	tglngne 29018	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
36	35	necomd 3010	. . . . . 6 ⊢ (𝜑 → 𝑍 ≠ 𝑋)
37	1, 17, 11, 6, 8, 9, 10, 16, 24	tgcgrcomlr 28947	. . . . . . 7 ⊢ (𝜑 → (𝑌 − 𝑋) = (𝑊 − 𝑍))
38	37	eqcomd 2766	. . . . . 6 ⊢ (𝜑 → (𝑊 − 𝑍) = (𝑌 − 𝑋))
39	1, 17, 11, 6, 9, 10, 16, 8, 25	tgcgrcomlr 28947	. . . . . 6 ⊢ (𝜑 → (𝑍 − 𝑌) = (𝑋 − 𝑊))
40	1, 2, 11, 6, 9, 16, 23, 28	colcom 29026	. . . . . 6 ⊢ (𝜑 → (𝑇 ∈ (𝑊𝐿𝑌) ∨ 𝑊 = 𝑌))
41	1, 2, 11, 6, 8, 10, 23, 27	colcom 29026	. . . . . 6 ⊢ (𝜑 → (𝑇 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
42	1, 17, 11, 2, 18, 6, 19, 16, 10, 9, 8, 23, 34, 36, 38, 39, 40, 41	symquadlem 29166	. . . . 5 ⊢ (𝜑 → 𝑊 = (((pInvG‘𝐺)‘𝑇)‘𝑌))
43	1, 17, 11, 6, 30, 9, 16, 18, 23	ismidb 29288	. . . . 5 ⊢ (𝜑 → (𝑊 = (((pInvG‘𝐺)‘𝑇)‘𝑌) ↔ (𝑌(midG‘𝐺)𝑊) = 𝑇))
44	42, 43	mpbid 235	. . . 4 ⊢ (𝜑 → (𝑌(midG‘𝐺)𝑊) = 𝑇)
45	32, 33, 44	3eqtr4d 2805	. . 3 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) = (𝑌(midG‘𝐺)𝑊))
46	1, 11, 2, 6, 8, 9, 10, 12	ncolne1 29098	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
47	1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 16, 45, 46	prlngmid2 29444	. 2 ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
48	34	orsild 1019	. . . . 5 ⊢ (𝜑 → ¬ 𝑊 ∈ (𝑍𝐿𝑌))
49	34	orsird 1020	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑍 = 𝑌)
50	49	neqned 2962	. . . . . 6 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
51	1, 11, 2, 6, 10, 9, 50	tglinecom 29108	. . . . 5 ⊢ (𝜑 → (𝑍𝐿𝑌) = (𝑌𝐿𝑍))
52	48, 51	neleqtrd 2882	. . . 4 ⊢ (𝜑 → ¬ 𝑊 ∈ (𝑌𝐿𝑍))
53	16, 52	eldifd 4050	. . 3 ⊢ (𝜑 → 𝑊 ∈ (𝑃 ∖ (𝑌𝐿𝑍)))
54	44, 32	eqtr4d 2798	. . 3 ⊢ (𝜑 → (𝑌(midG‘𝐺)𝑊) = (𝑍(midG‘𝐺)𝑋))
55	50	necomd 3010	. . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
56	1, 2, 3, 4, 5, 6, 7, 9, 10, 53, 8, 54, 55	prlngmid2 29444	. 2 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))
57	47, 56	jca 521	1 ⊢ (𝜑 → ((𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊) ∧ (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 Basecbs 17392 distcds 17442 TarskiGcstrkg 28894 TarskiG_Ecstrkge 28899 Itvcitv 28900 LineGclng 28901 pInvGcmir 29129 hlGcplng 29256 midGcmid 29282 parlnGcprlng 29419

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-oadd 8483 df-er 8720 df-map 8852 df-pm 8853 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-dju 9985 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-3 12409 df-n0 12610 df-xnn0 12683 df-z 12697 df-uz 12969 df-fz 13643 df-fzo 13792 df-hash 14478 df-word 14662 df-concat 14719 df-s1 14746 df-s2 15003 df-s3 15004 df-trkgc 28915 df-trkgb 28916 df-trkgcb 28917 df-trkgld 28919 df-trkg 28920 df-cgrg 28979 df-ismt 29001 df-leg 29051 df-hlg 29069 df-mir 29130 df-rag 29174 df-perpg 29176 df-hpg 29241 df-plng 29257 df-mid 29284 df-lmi 29285 df-cgra 29320 df-prlng 29420

This theorem is used by: (None)

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