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Theorem symquadlem 25497
Description: Lemma of the symetrial quadrilateral. The diagonals of quadrilaterals with congruent opposing sides intersect at their middle point. In Euclidean geometry, such quadrilaterals are called parallelograms, as opposing sides are parallel. However, this is not necessarily true in the case of absolute geometry. Lemma 7.21 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 6-Aug-2019.)
Hypotheses
Ref Expression
mirval.p 𝑃 = (Base‘𝐺)
mirval.d = (dist‘𝐺)
mirval.i 𝐼 = (Itv‘𝐺)
mirval.l 𝐿 = (LineG‘𝐺)
mirval.s 𝑆 = (pInvG‘𝐺)
mirval.g (𝜑𝐺 ∈ TarskiG)
symquadlem.m 𝑀 = (𝑆𝑋)
symquadlem.a (𝜑𝐴𝑃)
symquadlem.b (𝜑𝐵𝑃)
symquadlem.c (𝜑𝐶𝑃)
symquadlem.d (𝜑𝐷𝑃)
symquadlem.x (𝜑𝑋𝑃)
symquadlem.1 (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
symquadlem.2 (𝜑𝐵𝐷)
symquadlem.3 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
symquadlem.4 (𝜑 → (𝐵 𝐶) = (𝐷 𝐴))
symquadlem.5 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
symquadlem.6 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
Assertion
Ref Expression
symquadlem (𝜑𝐴 = (𝑀𝐶))

Proof of Theorem symquadlem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 symquadlem.1 . . . . . . . 8 (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
2 mirval.p . . . . . . . . . . 11 𝑃 = (Base‘𝐺)
3 mirval.l . . . . . . . . . . 11 𝐿 = (LineG‘𝐺)
4 mirval.i . . . . . . . . . . 11 𝐼 = (Itv‘𝐺)
5 mirval.g . . . . . . . . . . 11 (𝜑𝐺 ∈ TarskiG)
6 symquadlem.b . . . . . . . . . . 11 (𝜑𝐵𝑃)
7 symquadlem.a . . . . . . . . . . 11 (𝜑𝐴𝑃)
8 mirval.d . . . . . . . . . . . 12 = (dist‘𝐺)
92, 8, 4, 5, 6, 7tgbtwntriv2 25295 . . . . . . . . . . 11 (𝜑𝐴 ∈ (𝐵𝐼𝐴))
102, 3, 4, 5, 6, 7, 7, 9btwncolg1 25363 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1110adantr 481 . . . . . . . . 9 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
12 simpr 477 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1312oveq2d 6626 . . . . . . . . . . 11 ((𝜑𝐴 = 𝐶) → (𝐵𝐿𝐴) = (𝐵𝐿𝐶))
1413eleq2d 2684 . . . . . . . . . 10 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ↔ 𝐴 ∈ (𝐵𝐿𝐶)))
1512eqeq2d 2631 . . . . . . . . . 10 ((𝜑𝐴 = 𝐶) → (𝐵 = 𝐴𝐵 = 𝐶))
1614, 15orbi12d 745 . . . . . . . . 9 ((𝜑𝐴 = 𝐶) → ((𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴) ↔ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)))
1711, 16mpbid 222 . . . . . . . 8 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
181, 17mtand 690 . . . . . . 7 (𝜑 → ¬ 𝐴 = 𝐶)
1918neqned 2797 . . . . . 6 (𝜑𝐴𝐶)
2019ad2antrr 761 . . . . 5 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴𝐶)
2120necomd 2845 . . . 4 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶𝐴)
2221neneqd 2795 . . 3 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐶 = 𝐴)
23 mirval.s . . . . . 6 𝑆 = (pInvG‘𝐺)
245ad2antrr 761 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐺 ∈ TarskiG)
25 symquadlem.m . . . . . 6 𝑀 = (𝑆𝑋)
26 symquadlem.c . . . . . . 7 (𝜑𝐶𝑃)
2726ad2antrr 761 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶𝑃)
287ad2antrr 761 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴𝑃)
29 symquadlem.x . . . . . . 7 (𝜑𝑋𝑃)
3029ad2antrr 761 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋𝑃)
31 symquadlem.5 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
3231ad2antrr 761 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
332, 3, 4, 24, 28, 27, 30, 32colcom 25366 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
346ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐵𝑃)
35 symquadlem.d . . . . . . . . 9 (𝜑𝐷𝑃)
3635ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐷𝑃)
37 eqid 2621 . . . . . . . 8 (cgrG‘𝐺) = (cgrG‘𝐺)
38 simplr 791 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥𝑃)
39 symquadlem.6 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
402, 3, 4, 5, 6, 35, 29, 39colrot2 25368 . . . . . . . . . 10 (𝜑 → (𝐷 ∈ (𝑋𝐿𝐵) ∨ 𝑋 = 𝐵))
412, 3, 4, 5, 29, 6, 35, 40colcom 25366 . . . . . . . . 9 (𝜑 → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
4241ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
43 simpr 477 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
44 symquadlem.4 . . . . . . . . 9 (𝜑 → (𝐵 𝐶) = (𝐷 𝐴))
4544ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 𝐶) = (𝐷 𝐴))
46 symquadlem.3 . . . . . . . . . . 11 (𝜑 → (𝐴 𝐵) = (𝐶 𝐷))
472, 8, 4, 5, 7, 6, 26, 35, 46tgcgrcomlr 25288 . . . . . . . . . 10 (𝜑 → (𝐵 𝐴) = (𝐷 𝐶))
4847ad2antrr 761 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 𝐴) = (𝐷 𝐶))
4948eqcomd 2627 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 𝐶) = (𝐵 𝐴))
50 symquadlem.2 . . . . . . . . 9 (𝜑𝐵𝐷)
5150ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐵𝐷)
522, 3, 4, 24, 34, 36, 30, 37, 36, 34, 8, 27, 38, 28, 42, 43, 45, 49, 51tgfscgr 25376 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐶) = (𝑥 𝐴))
532, 3, 4, 5, 6, 26, 7, 1ncolcom 25369 . . . . . . . . . 10 (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
5453ad2antrr 761 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
5531orcomd 403 . . . . . . . . . . . 12 (𝜑 → (𝐴 = 𝐶𝑋 ∈ (𝐴𝐿𝐶)))
5655ord 392 . . . . . . . . . . 11 (𝜑 → (¬ 𝐴 = 𝐶𝑋 ∈ (𝐴𝐿𝐶)))
5718, 56mpd 15 . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐴𝐿𝐶))
5857ad2antrr 761 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐴𝐿𝐶))
5918ad2antrr 761 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐴 = 𝐶)
6045eqcomd 2627 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 𝐴) = (𝐵 𝐶))
612, 3, 4, 24, 34, 36, 30, 37, 36, 34, 8, 28, 38, 27, 42, 43, 48, 60, 51tgfscgr 25376 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐴) = (𝑥 𝐶))
622, 8, 4, 24, 30, 28, 38, 27, 61tgcgrcomlr 25288 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 𝑋) = (𝐶 𝑥))
632, 8, 4, 24, 27, 28axtgcgrrflx 25274 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 𝐴) = (𝐴 𝐶))
642, 8, 37, 24, 28, 30, 27, 27, 38, 28, 62, 52, 63trgcgr 25324 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐴𝑋𝐶”⟩(cgrG‘𝐺)⟨“𝐶𝑥𝐴”⟩)
652, 3, 4, 24, 28, 30, 27, 37, 27, 38, 28, 32, 64lnxfr 25374 . . . . . . . . . . . . 13 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
662, 3, 4, 24, 27, 28, 38, 65colcom 25366 . . . . . . . . . . . 12 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
6766orcomd 403 . . . . . . . . . . 11 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = 𝐶𝑥 ∈ (𝐴𝐿𝐶)))
6867ord 392 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (¬ 𝐴 = 𝐶𝑥 ∈ (𝐴𝐿𝐶)))
6959, 68mpd 15 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥 ∈ (𝐴𝐿𝐶))
7050neneqd 2795 . . . . . . . . . . 11 (𝜑 → ¬ 𝐵 = 𝐷)
7139orcomd 403 . . . . . . . . . . . 12 (𝜑 → (𝐵 = 𝐷𝑋 ∈ (𝐵𝐿𝐷)))
7271ord 392 . . . . . . . . . . 11 (𝜑 → (¬ 𝐵 = 𝐷𝑋 ∈ (𝐵𝐿𝐷)))
7370, 72mpd 15 . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐵𝐿𝐷))
7473ad2antrr 761 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐵𝐿𝐷))
7570ad2antrr 761 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐵 = 𝐷)
7639ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
772, 8, 4, 37, 24, 34, 36, 30, 36, 34, 38, 43cgr3swap23 25332 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝑋𝐷”⟩(cgrG‘𝐺)⟨“𝐷𝑥𝐵”⟩)
782, 3, 4, 24, 34, 30, 36, 37, 36, 38, 34, 76, 77lnxfr 25374 . . . . . . . . . . . . 13 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐷𝐿𝐵) ∨ 𝐷 = 𝐵))
792, 3, 4, 24, 36, 34, 38, 78colcom 25366 . . . . . . . . . . . 12 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
8079orcomd 403 . . . . . . . . . . 11 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 = 𝐷𝑥 ∈ (𝐵𝐿𝐷)))
8180ord 392 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (¬ 𝐵 = 𝐷𝑥 ∈ (𝐵𝐿𝐷)))
8275, 81mpd 15 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥 ∈ (𝐵𝐿𝐷))
832, 4, 3, 24, 28, 27, 34, 36, 54, 58, 69, 74, 82tglineinteq 25453 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 = 𝑥)
8483oveq1d 6625 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐴) = (𝑥 𝐴))
8552, 84eqtr4d 2658 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐶) = (𝑋 𝐴))
862, 8, 4, 3, 23, 24, 25, 27, 28, 30, 33, 85colmid 25496 . . . . 5 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = (𝑀𝐶) ∨ 𝐶 = 𝐴))
8786orcomd 403 . . . 4 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 = 𝐴𝐴 = (𝑀𝐶)))
8887ord 392 . . 3 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (¬ 𝐶 = 𝐴𝐴 = (𝑀𝐶)))
8922, 88mpd 15 . 2 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴 = (𝑀𝐶))
902, 8, 4, 5, 6, 35axtgcgrrflx 25274 . . 3 (𝜑 → (𝐵 𝐷) = (𝐷 𝐵))
912, 3, 4, 5, 6, 35, 29, 37, 35, 6, 8, 41, 90lnext 25375 . 2 (𝜑 → ∃𝑥𝑃 ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
9289, 91r19.29a 3072 1 (𝜑𝐴 = (𝑀𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4618  cfv 5852  (class class class)co 6610  ⟨“cs3 13531  Basecbs 15788  distcds 15878  TarskiGcstrkg 25242  Itvcitv 25248  LineGclng 25249  cgrGccgrg 25318  pInvGcmir 25460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-xnn0 11315  df-z 11329  df-uz 11639  df-fz 12276  df-fzo 12414  df-hash 13065  df-word 13245  df-concat 13247  df-s1 13248  df-s2 13537  df-s3 13538  df-trkgc 25260  df-trkgb 25261  df-trkgcb 25262  df-trkg 25265  df-cgrg 25319  df-mir 25461
This theorem is referenced by:  opphllem  25540
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