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Theorem prlngsymquadlem 29323

Description: Lemma for prlngsymquad 29324. (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)
prlngsymquad.2	⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
prlngsymquad.3	⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
prlngsymquad.4	⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))
prlngsymquadlem.t	⊢ 𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)

**Assertion**
Ref	Expression
prlngsymquadlem	⊢ (𝜑 → 𝑇 = 𝑊)

**Proof of Theorem prlngsymquadlem**
Step	Hyp	Ref	Expression
1		symquadprlng.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2760	. 2 ⊢ (Itv‘𝐺) = (Itv‘𝐺)
3		symquadprlng.l	. 2 ⊢ 𝐿 = (LineG‘𝐺)
4		symquadprlng.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		symquadprlng.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
6		symquadprlng.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7		symquadprlng.r	. . . . . 6 ⊢ ∥ = (parlnG‘𝐺)
8		prlngsymquad.4	. . . . . 6 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))
9	3, 7, 4, 8	prlngrcl2 29303	. . . . 5 ⊢ (𝜑 → (𝑊𝐿𝑋) ∈ ran 𝐿)
10	1, 2, 3, 4, 5, 6, 9	tglnne 28978	. . . 4 ⊢ (𝜑 → 𝑊 ≠ 𝑋)
11	1, 2, 3, 4, 5, 6, 10	tglinecom 28985	. . 3 ⊢ (𝜑 → (𝑊𝐿𝑋) = (𝑋𝐿𝑊))
12	11, 9	eqeltrrd 2861	. 2 ⊢ (𝜑 → (𝑋𝐿𝑊) ∈ ran 𝐿)
13		prlngsymquad.3	. . 3 ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
14	3, 7, 4, 13	prlngrcl2 29303	. 2 ⊢ (𝜑 → (𝑍𝐿𝑊) ∈ ran 𝐿)
15		symquadprlng.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
16		symquadprlng.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
17		prlngsymquad.2	. . . 4 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
18	1, 2, 3, 4, 6, 15, 16, 5, 17	tglineneq 28995	. . 3 ⊢ (𝜑 → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))
19	4	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝐺 ∈ TarskiG)
20	13	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
21		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))
22	3, 7, 19, 20, 21	prlngin0 29304	. . . . 5 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) = ∅)
23	1, 2, 3, 4, 6, 15, 16, 17	ncolne1 28975	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
24	1, 2, 3, 4, 6, 15, 23	tglinerflx1 28983	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿𝑌))
25	24	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ (𝑋𝐿𝑌))
26	10	necomd 3010	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ≠ 𝑊)
27	1, 2, 3, 4, 6, 5, 26	tglinerflx1 28983	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿𝑊))
28	27	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ (𝑋𝐿𝑊))
29		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → (𝑋𝐿𝑊) = (𝑍𝐿𝑊))
30	28, 29	eleqtrd 2862	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ (𝑍𝐿𝑊))
31	25, 30	elind 4146	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → 𝑋 ∈ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)))
32	31	ne0d 4288	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) ≠ ∅)
33	32	neneqd 2960	. . . . 5 ⊢ (((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) ∧ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) → ¬ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) = ∅)
34	22, 33	pm2.65da 829	. . . 4 ⊢ ((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) → ¬ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))
35		nne 2959	. . . 4 ⊢ (¬ (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊) ↔ (𝑋𝐿𝑌) = (𝑍𝐿𝑊))
36	34, 35	sylib 221	. . 3 ⊢ ((𝜑 ∧ (𝑋𝐿𝑊) = (𝑍𝐿𝑊)) → (𝑋𝐿𝑌) = (𝑍𝐿𝑊))
37	18, 36	mteqand 3046	. 2 ⊢ (𝜑 → (𝑋𝐿𝑊) ≠ (𝑍𝐿𝑊))
38		prlngsymquadlem.t	. . . . . 6 ⊢ 𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)
39		symquadprlng.d	. . . . . . 7 ⊢ − = (dist‘𝐺)
40		eqid 2760	. . . . . . 7 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
41	1, 3, 2, 4, 15, 16, 6, 17	ncoltgdim2 28910	. . . . . . . 8 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
42	1, 39, 2, 4, 41, 6, 16	midcl 29164	. . . . . . 7 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) ∈ 𝑃)
43		eqid 2760	. . . . . . 7 ⊢ ((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍)) = ((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))
44	1, 39, 2, 3, 40, 4, 42, 43, 15	mircl 29015	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) ∈ 𝑃)
45	38, 44	eqeltrid 2864	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
46	3, 7, 4, 8	prlngrcl1 29302	. . . . . . . . 9 ⊢ (𝜑 → (𝑌𝐿𝑍) ∈ ran 𝐿)
47	1, 2, 3, 4, 15, 16, 46	tglnne 28978	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
48	47	necomd 3010	. . . . . . 7 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
49	1, 40, 43, 4, 42, 16, 15	mirleqb 29048	. . . . . . . 8 ⊢ (𝜑 → (𝑍 = 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
50	49	necon3bid 2999	. . . . . . 7 ⊢ (𝜑 → (𝑍 ≠ 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
51	48, 50	mpbid 235	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌))
52		eqidd 2761	. . . . . . . . 9 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) = (𝑋(midG‘𝐺)𝑍))
53	1, 39, 2, 4, 41, 6, 16, 40, 42	ismidb 29165	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) ↔ (𝑋(midG‘𝐺)𝑍) = (𝑋(midG‘𝐺)𝑍)))
54	52, 53	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → 𝑍 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋))
55	54	eqcomd 2766	. . . . . . 7 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) = 𝑍)
56	1, 39, 2, 3, 40, 4, 42, 43, 6, 55	mircom 29017	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑍) = 𝑋)
57	38	eqcomi 2769	. . . . . . 7 ⊢ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) = 𝑇
58	57	a1i 11	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) = 𝑇)
59	51, 56, 58	3netr3d 3031	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑇)
60	1, 2, 3, 4, 6, 45, 59	tglinerflx2 28984	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑇))
61	1, 40, 43, 4, 42, 6, 15	mirleqb 29048	. . . . . . . 8 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
62	61	necon3bid 2999	. . . . . . 7 ⊢ (𝜑 → (𝑋 ≠ 𝑌 ↔ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)))
63	23, 62	mpbid 235	. . . . . 6 ⊢ (𝜑 → (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑋) ≠ (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌))
64	63, 55, 58	3netr3d 3031	. . . . 5 ⊢ (𝜑 → 𝑍 ≠ 𝑇)
65	1, 2, 3, 4, 16, 45, 64	tglinerflx2 28984	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (𝑍𝐿𝑇))
66	60, 65	elind 4146	. . 3 ⊢ (𝜑 → 𝑇 ∈ ((𝑋𝐿𝑇) ∩ (𝑍𝐿𝑇)))
67		symquadprlng.1	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG_E)
68	1, 2, 3, 4, 15, 16, 47	tglinecom 28985	. . . . . 6 ⊢ (𝜑 → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
69		eqid 2760	. . . . . . 7 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
70		eqid 2760	. . . . . . 7 ⊢ (midG‘𝐺) = (midG‘𝐺)
71	1, 3, 2, 4, 15, 16, 6, 17	ncolcom 28906	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
72	71	orsild 1019	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑍𝐿𝑌))
73	6, 72	eldifd 3910	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))
74	1, 39, 2, 4, 41, 16, 6	midcom 29169	. . . . . . . 8 ⊢ (𝜑 → (𝑍(midG‘𝐺)𝑋) = (𝑋(midG‘𝐺)𝑍))
75	1, 39, 2, 4, 41, 15, 45, 40, 42	ismidb 29165	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) ↔ (𝑌(midG‘𝐺)𝑇) = (𝑋(midG‘𝐺)𝑍)))
76	38, 75	mpbii 236	. . . . . . . . 9 ⊢ (𝜑 → (𝑌(midG‘𝐺)𝑇) = (𝑋(midG‘𝐺)𝑍))
77	76	eqcomd 2766	. . . . . . . 8 ⊢ (𝜑 → (𝑋(midG‘𝐺)𝑍) = (𝑌(midG‘𝐺)𝑇))
78	74, 77	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → (𝑍(midG‘𝐺)𝑋) = (𝑌(midG‘𝐺)𝑇))
79	1, 3, 69, 7, 70, 4, 67, 16, 15, 73, 45, 78, 48	prlngmid2 29321	. . . . . 6 ⊢ (𝜑 → (𝑍𝐿𝑌) ∥ (𝑋𝐿𝑇))
80	68, 79	eqbrtrd 5127	. . . . 5 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑋𝐿𝑇))
81	8, 11	breqtrd 5131	. . . . 5 ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑋𝐿𝑊))
82	1, 2, 3, 4, 6, 45, 59	tglinerflx1 28983	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿𝑇))
83	1, 7, 4, 67, 80, 81, 82, 27	prlngeq 29317	. . . 4 ⊢ (𝜑 → (𝑋𝐿𝑇) = (𝑋𝐿𝑊))
84	1, 3, 2, 4, 15, 16, 6, 17	ncolrot2 28908	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
85	84	orsild 1019	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑋𝐿𝑌))
86	16, 85	eldifd 3910	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))
87	1, 3, 69, 7, 70, 4, 67, 6, 15, 86, 45, 77, 23	prlngmid2 29321	. . . . 5 ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑇))
88	1, 2, 3, 4, 16, 45, 64	tglinerflx1 28983	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑍𝐿𝑇))
89	1, 2, 3, 4, 16, 5, 14	tglnne 28978	. . . . . 6 ⊢ (𝜑 → 𝑍 ≠ 𝑊)
90	1, 2, 3, 4, 16, 5, 89	tglinerflx1 28983	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑍𝐿𝑊))
91	1, 7, 4, 67, 87, 13, 88, 90	prlngeq 29317	. . . 4 ⊢ (𝜑 → (𝑍𝐿𝑇) = (𝑍𝐿𝑊))
92	83, 91	ineq12d 4167	. . 3 ⊢ (𝜑 → ((𝑋𝐿𝑇) ∩ (𝑍𝐿𝑇)) = ((𝑋𝐿𝑊) ∩ (𝑍𝐿𝑊)))
93	66, 92	eleqtrd 2862	. 2 ⊢ (𝜑 → 𝑇 ∈ ((𝑋𝐿𝑊) ∩ (𝑍𝐿𝑊)))
94	1, 2, 3, 4, 6, 5, 26	tglinerflx2 28984	. . 3 ⊢ (𝜑 → 𝑊 ∈ (𝑋𝐿𝑊))
95	1, 2, 3, 4, 16, 5, 89	tglinerflx2 28984	. . 3 ⊢ (𝜑 → 𝑊 ∈ (𝑍𝐿𝑊))
96	94, 95	elind 4146	. 2 ⊢ (𝜑 → 𝑊 ∈ ((𝑋𝐿𝑊) ∩ (𝑍𝐿𝑊)))
97	1, 2, 3, 4, 12, 14, 37, 93, 96	tglineineq 28993	1 ⊢ (𝜑 → 𝑇 = 𝑊)

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 2955 ∩ cin 3898 ∅c0 4279 class class class wbr 5103 ran crn 5656 ‘cfv 6533 (class class class)co 7414 Basecbs 17304 distcds 17354 TarskiGcstrkg 28771 TarskiG_Ecstrkge 28776 Itvcitv 28777 LineGclng 28778 pInvGcmir 29006 hlGcplng 29133 midGcmid 29159 parlnGcprlng 29296

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 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

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 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 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 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-oadd 8462 df-er 8699 df-map 8831 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9909 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-xnn0 12605 df-z 12619 df-uz 12891 df-fz 13565 df-fzo 13713 df-hash 14398 df-word 14582 df-concat 14639 df-s1 14666 df-s2 14922 df-s3 14923 df-trkgc 28792 df-trkgb 28793 df-trkgcb 28794 df-trkge 28795 df-trkgld 28796 df-trkg 28797 df-cgrg 28856 df-ismt 28878 df-leg 28928 df-hlg 28946 df-mir 29007 df-rag 29051 df-perpg 29053 df-hpg 29118 df-plng 29134 df-mid 29161 df-lmi 29162 df-cgra 29197 df-prlng 29297

This theorem is used by: prlngsymquad 29324 prlngsymquadopp 29325

W3C validator