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Theorem symquadlem 26489
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 26287 . . . . . . . . . . 11 (𝜑𝐴 ∈ (𝐵𝐼𝐴))
102, 3, 4, 5, 6, 7, 7, 9btwncolg1 26355 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1110adantr 484 . . . . . . . . 9 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
12 simpr 488 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐶) → 𝐴 = 𝐶)
1312oveq2d 7165 . . . . . . . . . . 11 ((𝜑𝐴 = 𝐶) → (𝐵𝐿𝐴) = (𝐵𝐿𝐶))
1413eleq2d 2901 . . . . . . . . . 10 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐴) ↔ 𝐴 ∈ (𝐵𝐿𝐶)))
1512eqeq2d 2835 . . . . . . . . . 10 ((𝜑𝐴 = 𝐶) → (𝐵 = 𝐴𝐵 = 𝐶))
1614, 15orbi12d 916 . . . . . . . . 9 ((𝜑𝐴 = 𝐶) → ((𝐴 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴) ↔ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)))
1711, 16mpbid 235 . . . . . . . 8 ((𝜑𝐴 = 𝐶) → (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
181, 17mtand 815 . . . . . . 7 (𝜑 → ¬ 𝐴 = 𝐶)
1918neqned 3021 . . . . . 6 (𝜑𝐴𝐶)
2019ad2antrr 725 . . . . 5 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴𝐶)
2120necomd 3069 . . . 4 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶𝐴)
2221neneqd 3019 . . 3 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐶 = 𝐴)
23 mirval.s . . . . . 6 𝑆 = (pInvG‘𝐺)
245ad2antrr 725 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐺 ∈ TarskiG)
25 symquadlem.m . . . . . 6 𝑀 = (𝑆𝑋)
26 symquadlem.c . . . . . . 7 (𝜑𝐶𝑃)
2726ad2antrr 725 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐶𝑃)
287ad2antrr 725 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐴𝑃)
29 symquadlem.x . . . . . . 7 (𝜑𝑋𝑃)
3029ad2antrr 725 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋𝑃)
31 symquadlem.5 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
3231ad2antrr 725 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
332, 3, 4, 24, 28, 27, 30, 32colcom 26358 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
346ad2antrr 725 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐵𝑃)
35 symquadlem.d . . . . . . . . 9 (𝜑𝐷𝑃)
3635ad2antrr 725 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝐷𝑃)
37 eqid 2824 . . . . . . . 8 (cgrG‘𝐺) = (cgrG‘𝐺)
38 simplr 768 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥𝑃)
39 symquadlem.6 . . . . . . . . . . 11 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
402, 3, 4, 5, 6, 35, 29, 39colrot2 26360 . . . . . . . . . 10 (𝜑 → (𝐷 ∈ (𝑋𝐿𝐵) ∨ 𝑋 = 𝐵))
412, 3, 4, 5, 29, 6, 35, 40colcom 26358 . . . . . . . . 9 (𝜑 → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
4241ad2antrr 725 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 ∈ (𝐵𝐿𝑋) ∨ 𝐵 = 𝑋))
43 simpr 488 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
44 symquadlem.4 . . . . . . . . 9 (𝜑 → (𝐵 𝐶) = (𝐷 𝐴))
4544ad2antrr 725 . . . . . . . 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 26280 . . . . . . . . . 10 (𝜑 → (𝐵 𝐴) = (𝐷 𝐶))
4847ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 𝐴) = (𝐷 𝐶))
4948eqcomd 2830 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐷 𝐶) = (𝐵 𝐴))
50 symquadlem.2 . . . . . . . . 9 (𝜑𝐵𝐷)
5150ad2antrr 725 . . . . . . . 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 26368 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐶) = (𝑥 𝐴))
532, 3, 4, 5, 6, 26, 7, 1ncolcom 26361 . . . . . . . . . 10 (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
5453ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ (𝐴 ∈ (𝐶𝐿𝐵) ∨ 𝐶 = 𝐵))
5531orcomd 868 . . . . . . . . . . . 12 (𝜑 → (𝐴 = 𝐶𝑋 ∈ (𝐴𝐿𝐶)))
5655ord 861 . . . . . . . . . . 11 (𝜑 → (¬ 𝐴 = 𝐶𝑋 ∈ (𝐴𝐿𝐶)))
5718, 56mpd 15 . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐴𝐿𝐶))
5857ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐴𝐿𝐶))
5918ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐴 = 𝐶)
6045eqcomd 2830 . . . . . . . . . . . . . . . . 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 26368 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐴) = (𝑥 𝐶))
622, 8, 4, 24, 30, 28, 38, 27, 61tgcgrcomlr 26280 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 𝑋) = (𝐶 𝑥))
632, 8, 4, 24, 27, 28axtgcgrrflx 26262 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 𝐴) = (𝐴 𝐶))
642, 8, 37, 24, 28, 30, 27, 27, 38, 28, 62, 52, 63trgcgr 26316 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐴𝑋𝐶”⟩(cgrG‘𝐺)⟨“𝐶𝑥𝐴”⟩)
652, 3, 4, 24, 28, 30, 27, 37, 27, 38, 28, 32, 64lnxfr 26366 . . . . . . . . . . . . 13 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐶𝐿𝐴) ∨ 𝐶 = 𝐴))
662, 3, 4, 24, 27, 28, 38, 65colcom 26358 . . . . . . . . . . . 12 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
6766orcomd 868 . . . . . . . . . . 11 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = 𝐶𝑥 ∈ (𝐴𝐿𝐶)))
6867ord 861 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (¬ 𝐴 = 𝐶𝑥 ∈ (𝐴𝐿𝐶)))
6959, 68mpd 15 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑥 ∈ (𝐴𝐿𝐶))
7050neneqd 3019 . . . . . . . . . . 11 (𝜑 → ¬ 𝐵 = 𝐷)
7139orcomd 868 . . . . . . . . . . . 12 (𝜑 → (𝐵 = 𝐷𝑋 ∈ (𝐵𝐿𝐷)))
7271ord 861 . . . . . . . . . . 11 (𝜑 → (¬ 𝐵 = 𝐷𝑋 ∈ (𝐵𝐿𝐷)))
7370, 72mpd 15 . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐵𝐿𝐷))
7473ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 ∈ (𝐵𝐿𝐷))
7570ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ¬ 𝐵 = 𝐷)
7639ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
772, 8, 4, 37, 24, 34, 36, 30, 36, 34, 38, 43cgr3swap23 26324 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → ⟨“𝐵𝑋𝐷”⟩(cgrG‘𝐺)⟨“𝐷𝑥𝐵”⟩)
782, 3, 4, 24, 34, 30, 36, 37, 36, 38, 34, 76, 77lnxfr 26366 . . . . . . . . . . . . 13 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐷𝐿𝐵) ∨ 𝐷 = 𝐵))
792, 3, 4, 24, 36, 34, 38, 78colcom 26358 . . . . . . . . . . . 12 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑥 ∈ (𝐵𝐿𝐷) ∨ 𝐵 = 𝐷))
8079orcomd 868 . . . . . . . . . . 11 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐵 = 𝐷𝑥 ∈ (𝐵𝐿𝐷)))
8180ord 861 . . . . . . . . . 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 26445 . . . . . . . 8 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → 𝑋 = 𝑥)
8483oveq1d 7164 . . . . . . 7 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐴) = (𝑥 𝐴))
8552, 84eqtr4d 2862 . . . . . 6 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝑋 𝐶) = (𝑋 𝐴))
862, 8, 4, 3, 23, 24, 25, 27, 28, 30, 33, 85colmid 26488 . . . . 5 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐴 = (𝑀𝐶) ∨ 𝐶 = 𝐴))
8786orcomd 868 . . . 4 (((𝜑𝑥𝑃) ∧ ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩) → (𝐶 = 𝐴𝐴 = (𝑀𝐶)))
8887ord 861 . . 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 26262 . . 3 (𝜑 → (𝐵 𝐷) = (𝐷 𝐵))
912, 3, 4, 5, 6, 35, 29, 37, 35, 6, 8, 41, 90lnext 26367 . 2 (𝜑 → ∃𝑥𝑃 ⟨“𝐵𝐷𝑋”⟩(cgrG‘𝐺)⟨“𝐷𝐵𝑥”⟩)
9289, 91r19.29a 3281 1 (𝜑𝐴 = (𝑀𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  (class class class)co 7149  ⟨“cs3 14204  Basecbs 16483  distcds 16574  TarskiGcstrkg 26230  Itvcitv 26236  LineGclng 26237  cgrGccgrg 26310  pInvGcmir 26452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-xnn0 11965  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-hash 13696  df-word 13867  df-concat 13923  df-s1 13950  df-s2 14210  df-s3 14211  df-trkgc 26248  df-trkgb 26249  df-trkgcb 26250  df-trkg 26253  df-cgrg 26311  df-mir 26453
This theorem is referenced by:  opphllem  26535
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