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Theorem symquadprlng 29422

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 2761	. . 3 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
4		symquadprlng.r	. . 3 ⊢ ∥ = (parlnG‘𝐺)
5		eqid 2761	. . 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 2761	. . . . . 6 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
12		symquadprlng.4	. . . . . 6 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
13	1, 2, 11, 6, 9, 10, 8, 12	ncolrot2 29008	. . . . 5 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
14	13	orsild 1019	. . . 4 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
15	10, 14	eldifd 3910	. . 3 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
16		symquadprlng.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
17		symquadprlng.d	. . . . . 6 ⊢ − = (dist‘𝐺)
18		eqid 2761	. . . . . 6 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
19		eqid 2761	. . . . . 6 ⊢ ((pInvG‘𝐺)‘𝑇) = ((pInvG‘𝐺)‘𝑇)
20		symquadprlng.5	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑊)
21	1, 11, 2, 6, 9, 16, 20	tgelrnln 29080	. . . . . . 7 ⊢ (𝜑 → (𝑌𝐿𝑊) ∈ ran 𝐿)
22		symquadprlng.7	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊))
23	1, 2, 11, 6, 21, 22	tglnpt 28994	. . . . . 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 29143	. . . . 5 ⊢ (𝜑 → 𝑋 = (((pInvG‘𝐺)‘𝑇)‘𝑍))
30	1, 2, 11, 6, 9, 10, 8, 12	ncoltgdim2 29010	. . . . . 6 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
31	1, 17, 11, 6, 30, 10, 8, 18, 23	ismidb 29265	. . . . 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 29269	. . . 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 29147	. . . . . 6 ⊢ (𝜑 → ¬ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
35	1, 2, 11, 6, 8, 10, 26	tglngne 28995	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
36	35	necomd 3011	. . . . . 6 ⊢ (𝜑 → 𝑍 ≠ 𝑋)
37	1, 17, 11, 6, 8, 9, 10, 16, 24	tgcgrcomlr 28924	. . . . . . 7 ⊢ (𝜑 → (𝑌 − 𝑋) = (𝑊 − 𝑍))
38	37	eqcomd 2767	. . . . . 6 ⊢ (𝜑 → (𝑊 − 𝑍) = (𝑌 − 𝑋))
39	1, 17, 11, 6, 9, 10, 16, 8, 25	tgcgrcomlr 28924	. . . . . 6 ⊢ (𝜑 → (𝑍 − 𝑌) = (𝑋 − 𝑊))
40	1, 2, 11, 6, 9, 16, 23, 28	colcom 29003	. . . . . 6 ⊢ (𝜑 → (𝑇 ∈ (𝑊𝐿𝑌) ∨ 𝑊 = 𝑌))
41	1, 2, 11, 6, 8, 10, 23, 27	colcom 29003	. . . . . 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 29143	. . . . 5 ⊢ (𝜑 → 𝑊 = (((pInvG‘𝐺)‘𝑇)‘𝑌))
43	1, 17, 11, 6, 30, 9, 16, 18, 23	ismidb 29265	. . . . 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 2806	. . 3 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) = (𝑌(midG‘𝐺)𝑊))
46	1, 11, 2, 6, 8, 9, 10, 12	ncolne1 29075	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
47	1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 16, 45, 46	prlngmid2 29421	. 2 ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
48	34	orsild 1019	. . . . 5 ⊢ (𝜑 → ¬ 𝑊 ∈ (𝑍𝐿𝑌))
49	34	orsird 1020	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑍 = 𝑌)
50	49	neqned 2963	. . . . . 6 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
51	1, 11, 2, 6, 10, 9, 50	tglinecom 29085	. . . . 5 ⊢ (𝜑 → (𝑍𝐿𝑌) = (𝑌𝐿𝑍))
52	48, 51	neleqtrd 2883	. . . 4 ⊢ (𝜑 → ¬ 𝑊 ∈ (𝑌𝐿𝑍))
53	16, 52	eldifd 3910	. . 3 ⊢ (𝜑 → 𝑊 ∈ (𝑃 ∖ (𝑌𝐿𝑍)))
54	44, 32	eqtr4d 2799	. . 3 ⊢ (𝜑 → (𝑌(midG‘𝐺)𝑊) = (𝑍(midG‘𝐺)𝑋))
55	50	necomd 3011	. . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
56	1, 2, 3, 4, 5, 6, 7, 9, 10, 53, 8, 54, 55	prlngmid2 29421	. 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 1570 ∈ wcel 2145 ≠ wne 2956 class class class wbr 5103 ‘cfv 6531 (class class class)co 7412 Basecbs 17367 distcds 17417 TarskiGcstrkg 28871 TarskiG_Ecstrkge 28876 Itvcitv 28877 LineGclng 28878 pInvGcmir 29106 hlGcplng 29233 midGcmid 29259 parlnGcprlng 29396

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-oadd 8464 df-er 8701 df-map 8833 df-pm 8834 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9963 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-n0 12588 df-xnn0 12661 df-z 12675 df-uz 12947 df-fz 13621 df-fzo 13769 df-hash 14455 df-word 14639 df-concat 14696 df-s1 14723 df-s2 14979 df-s3 14980 df-trkgc 28892 df-trkgb 28893 df-trkgcb 28894 df-trkgld 28896 df-trkg 28897 df-cgrg 28956 df-ismt 28978 df-leg 29028 df-hlg 29046 df-mir 29107 df-rag 29151 df-perpg 29153 df-hpg 29218 df-plng 29234 df-mid 29261 df-lmi 29262 df-cgra 29297 df-prlng 29397

This theorem is used by: (None)

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