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Theorem symquadlem 28565
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 28363 . . . . . . . . . . 11 (𝜑𝐴 ∈ (𝐵𝐼𝐴))
102, 3, 4, 5, 6, 7, 7, 9btwncolg1 28431 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1110adantr 479 . . . . . . . . 9 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
12 simpr 483 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1312oveq2d 7435 . . . . . . . . . . 11 ((𝜑𝐴 = 𝐶) → (𝐵𝐿𝐴) = (𝐵𝐿𝐶))
1413eleq2d 2811 . . . . . . . . . 10 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ↔ 𝐴 ∈ (𝐵𝐿𝐶)))
1512eqeq2d 2736 . . . . . . . . . 10 ((𝜑𝐴 = 𝐶) → (𝐵 = 𝐴𝐵 = 𝐶))
1614, 15orbi12d 916 . . . . . . . . 9 ((𝜑𝐴 = 𝐶) → ((𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴) ↔ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)))
1711, 16mpbid 231 . . . . . . . 8 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
181, 17mtand 814 . . . . . . 7 (𝜑 → ¬ 𝐴 = 𝐶)
1918neqned 2936 . . . . . 6 (𝜑𝐴𝐶)
2019ad2antrr 724 . . . . 5 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴𝐶)
2120necomd 2985 . . . 4 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶𝐴)
2221neneqd 2934 . . 3 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐶 = 𝐴)
23 mirval.s . . . . . 6 𝑆 = (pInvG‘𝐺)
245ad2antrr 724 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐺 ∈ TarskiG)
25 symquadlem.m . . . . . 6 𝑀 = (𝑆𝑋)
26 symquadlem.c . . . . . . 7 (𝜑𝐶𝑃)
2726ad2antrr 724 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶𝑃)
287ad2antrr 724 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴𝑃)
29 symquadlem.x . . . . . . 7 (𝜑𝑋𝑃)
3029ad2antrr 724 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋𝑃)
31 symquadlem.5 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
3231ad2antrr 724 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
332, 3, 4, 24, 28, 27, 30, 32colcom 28434 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
346ad2antrr 724 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐵𝑃)
35 symquadlem.d . . . . . . . . 9 (𝜑𝐷𝑃)
3635ad2antrr 724 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐷𝑃)
37 eqid 2725 . . . . . . . 8 (cgrG‘𝐺) = (cgrG‘𝐺)
38 simplr 767 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥𝑃)
39 symquadlem.6 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
402, 3, 4, 5, 6, 35, 29, 39colrot2 28436 . . . . . . . . . 10 (𝜑 → (𝐷 ∈ (𝑋𝐿𝐵) ∨ 𝑋 = 𝐵))
412, 3, 4, 5, 29, 6, 35, 40colcom 28434 . . . . . . . . 9 (𝜑 → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
4241ad2antrr 724 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
43 simpr 483 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
44 symquadlem.4 . . . . . . . . 9 (𝜑 → (𝐵 𝐶) = (𝐷 𝐴))
4544ad2antrr 724 . . . . . . . 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 28356 . . . . . . . . . 10 (𝜑 → (𝐵 𝐴) = (𝐷 𝐶))
4847ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 𝐴) = (𝐷 𝐶))
4948eqcomd 2731 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 𝐶) = (𝐵 𝐴))
50 symquadlem.2 . . . . . . . . 9 (𝜑𝐵𝐷)
5150ad2antrr 724 . . . . . . . 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 28444 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐶) = (𝑥 𝐴))
532, 3, 4, 5, 6, 26, 7, 1ncolcom 28437 . . . . . . . . . 10 (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
5453ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
5531orcomd 869 . . . . . . . . . . . 12 (𝜑 → (𝐴 = 𝐶𝑋 ∈ (𝐴𝐿𝐶)))
5655ord 862 . . . . . . . . . . 11 (𝜑 → (¬ 𝐴 = 𝐶𝑋 ∈ (𝐴𝐿𝐶)))
5718, 56mpd 15 . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐴𝐿𝐶))
5857ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐴𝐿𝐶))
5918ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐴 = 𝐶)
6045eqcomd 2731 . . . . . . . . . . . . . . . . 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 28444 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐴) = (𝑥 𝐶))
622, 8, 4, 24, 30, 28, 38, 27, 61tgcgrcomlr 28356 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 𝑋) = (𝐶 𝑥))
632, 8, 4, 24, 27, 28axtgcgrrflx 28338 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 𝐴) = (𝐴 𝐶))
642, 8, 37, 24, 28, 30, 27, 27, 38, 28, 62, 52, 63trgcgr 28392 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐴𝑋𝐶”⟩(cgrG‘𝐺)⟨“𝐶𝑥𝐴”⟩)
652, 3, 4, 24, 28, 30, 27, 37, 27, 38, 28, 32, 64lnxfr 28442 . . . . . . . . . . . . 13 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
662, 3, 4, 24, 27, 28, 38, 65colcom 28434 . . . . . . . . . . . 12 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
6766orcomd 869 . . . . . . . . . . 11 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = 𝐶𝑥 ∈ (𝐴𝐿𝐶)))
6867ord 862 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (¬ 𝐴 = 𝐶𝑥 ∈ (𝐴𝐿𝐶)))
6959, 68mpd 15 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥 ∈ (𝐴𝐿𝐶))
7050neneqd 2934 . . . . . . . . . . 11 (𝜑 → ¬ 𝐵 = 𝐷)
7139orcomd 869 . . . . . . . . . . . 12 (𝜑 → (𝐵 = 𝐷𝑋 ∈ (𝐵𝐿𝐷)))
7271ord 862 . . . . . . . . . . 11 (𝜑 → (¬ 𝐵 = 𝐷𝑋 ∈ (𝐵𝐿𝐷)))
7370, 72mpd 15 . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐵𝐿𝐷))
7473ad2antrr 724 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐵𝐿𝐷))
7570ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐵 = 𝐷)
7639ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
772, 8, 4, 37, 24, 34, 36, 30, 36, 34, 38, 43cgr3swap23 28400 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝑋𝐷”⟩(cgrG‘𝐺)⟨“𝐷𝑥𝐵”⟩)
782, 3, 4, 24, 34, 30, 36, 37, 36, 38, 34, 76, 77lnxfr 28442 . . . . . . . . . . . . 13 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐷𝐿𝐵) ∨ 𝐷 = 𝐵))
792, 3, 4, 24, 36, 34, 38, 78colcom 28434 . . . . . . . . . . . 12 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
8079orcomd 869 . . . . . . . . . . 11 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 = 𝐷𝑥 ∈ (𝐵𝐿𝐷)))
8180ord 862 . . . . . . . . . 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 28521 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 = 𝑥)
8483oveq1d 7434 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐴) = (𝑥 𝐴))
8552, 84eqtr4d 2768 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐶) = (𝑋 𝐴))
862, 8, 4, 3, 23, 24, 25, 27, 28, 30, 33, 85colmid 28564 . . . . 5 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = (𝑀𝐶) ∨ 𝐶 = 𝐴))
8786orcomd 869 . . . 4 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 = 𝐴𝐴 = (𝑀𝐶)))
8887ord 862 . . 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 28338 . . 3 (𝜑 → (𝐵 𝐷) = (𝐷 𝐵))
912, 3, 4, 5, 6, 35, 29, 37, 35, 6, 8, 41, 90lnext 28443 . 2 (𝜑 → ∃𝑥𝑃 ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
9289, 91r19.29a 3151 1 (𝜑𝐴 = (𝑀𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 845   = wceq 1533  wcel 2098  wne 2929   class class class wbr 5149  cfv 6549  (class class class)co 7419  ⟨“cs3 14829  Basecbs 17183  distcds 17245  TarskiGcstrkg 28303  Itvcitv 28309  LineGclng 28310  cgrGccgrg 28386  pInvGcmir 28528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9926  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-xnn0 12578  df-z 12592  df-uz 12856  df-fz 13520  df-fzo 13663  df-hash 14326  df-word 14501  df-concat 14557  df-s1 14582  df-s2 14835  df-s3 14836  df-trkgc 28324  df-trkgb 28325  df-trkgcb 28326  df-trkg 28329  df-cgrg 28387  df-mir 28529
This theorem is referenced by:  opphllem  28611
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