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Theorem xmulasslem 11987
Description: Lemma for xmulass 11989. (Contributed by Mario Carneiro, 20-Aug-2015.)
Hypotheses
Ref Expression
xmulasslem.1 (𝑥 = 𝐷 → (𝜓𝑋 = 𝑌))
xmulasslem.2 (𝑥 = -𝑒𝐷 → (𝜓𝐸 = 𝐹))
xmulasslem.x (𝜑𝑋 ∈ ℝ*)
xmulasslem.y (𝜑𝑌 ∈ ℝ*)
xmulasslem.d (𝜑𝐷 ∈ ℝ*)
xmulasslem.ps ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝜓)
xmulasslem.0 (𝜑 → (𝑥 = 0 → 𝜓))
xmulasslem.e (𝜑𝐸 = -𝑒𝑋)
xmulasslem.f (𝜑𝐹 = -𝑒𝑌)
Assertion
Ref Expression
xmulasslem (𝜑𝑋 = 𝑌)
Distinct variable groups:   𝑥,𝐷   𝑥,𝐸   𝑥,𝐹   𝜑,𝑥   𝑥,𝑋   𝑥,𝑌
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem xmulasslem
StepHypRef Expression
1 xmulasslem.d . . 3 (𝜑𝐷 ∈ ℝ*)
2 0xr 9965 . . 3 0 ∈ ℝ*
3 xrltso 11850 . . . 4 < Or ℝ*
4 solin 4982 . . . 4 (( < Or ℝ* ∧ (𝐷 ∈ ℝ* ∧ 0 ∈ ℝ*)) → (𝐷 < 0 ∨ 𝐷 = 0 ∨ 0 < 𝐷))
53, 4mpan 702 . . 3 ((𝐷 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐷 < 0 ∨ 𝐷 = 0 ∨ 0 < 𝐷))
61, 2, 5sylancl 693 . 2 (𝜑 → (𝐷 < 0 ∨ 𝐷 = 0 ∨ 0 < 𝐷))
7 xlt0neg1 11924 . . . . . 6 (𝐷 ∈ ℝ* → (𝐷 < 0 ↔ 0 < -𝑒𝐷))
81, 7syl 17 . . . . 5 (𝜑 → (𝐷 < 0 ↔ 0 < -𝑒𝐷))
9 xnegcl 11918 . . . . . . 7 (𝐷 ∈ ℝ* → -𝑒𝐷 ∈ ℝ*)
101, 9syl 17 . . . . . 6 (𝜑 → -𝑒𝐷 ∈ ℝ*)
11 breq2 4587 . . . . . . . . 9 (𝑥 = -𝑒𝐷 → (0 < 𝑥 ↔ 0 < -𝑒𝐷))
12 xmulasslem.2 . . . . . . . . 9 (𝑥 = -𝑒𝐷 → (𝜓𝐸 = 𝐹))
1311, 12imbi12d 333 . . . . . . . 8 (𝑥 = -𝑒𝐷 → ((0 < 𝑥𝜓) ↔ (0 < -𝑒𝐷𝐸 = 𝐹)))
1413imbi2d 329 . . . . . . 7 (𝑥 = -𝑒𝐷 → ((𝜑 → (0 < 𝑥𝜓)) ↔ (𝜑 → (0 < -𝑒𝐷𝐸 = 𝐹))))
15 xmulasslem.ps . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ 0 < 𝑥)) → 𝜓)
1615exp32 629 . . . . . . . 8 (𝜑 → (𝑥 ∈ ℝ* → (0 < 𝑥𝜓)))
1716com12 32 . . . . . . 7 (𝑥 ∈ ℝ* → (𝜑 → (0 < 𝑥𝜓)))
1814, 17vtoclga 3245 . . . . . 6 (-𝑒𝐷 ∈ ℝ* → (𝜑 → (0 < -𝑒𝐷𝐸 = 𝐹)))
1910, 18mpcom 37 . . . . 5 (𝜑 → (0 < -𝑒𝐷𝐸 = 𝐹))
208, 19sylbid 229 . . . 4 (𝜑 → (𝐷 < 0 → 𝐸 = 𝐹))
21 xmulasslem.e . . . . . 6 (𝜑𝐸 = -𝑒𝑋)
22 xmulasslem.f . . . . . 6 (𝜑𝐹 = -𝑒𝑌)
2321, 22eqeq12d 2625 . . . . 5 (𝜑 → (𝐸 = 𝐹 ↔ -𝑒𝑋 = -𝑒𝑌))
24 xmulasslem.x . . . . . 6 (𝜑𝑋 ∈ ℝ*)
25 xmulasslem.y . . . . . 6 (𝜑𝑌 ∈ ℝ*)
26 xneg11 11920 . . . . . 6 ((𝑋 ∈ ℝ*𝑌 ∈ ℝ*) → (-𝑒𝑋 = -𝑒𝑌𝑋 = 𝑌))
2724, 25, 26syl2anc 691 . . . . 5 (𝜑 → (-𝑒𝑋 = -𝑒𝑌𝑋 = 𝑌))
2823, 27bitrd 267 . . . 4 (𝜑 → (𝐸 = 𝐹𝑋 = 𝑌))
2920, 28sylibd 228 . . 3 (𝜑 → (𝐷 < 0 → 𝑋 = 𝑌))
30 eqeq1 2614 . . . . . . 7 (𝑥 = 𝐷 → (𝑥 = 0 ↔ 𝐷 = 0))
31 xmulasslem.1 . . . . . . 7 (𝑥 = 𝐷 → (𝜓𝑋 = 𝑌))
3230, 31imbi12d 333 . . . . . 6 (𝑥 = 𝐷 → ((𝑥 = 0 → 𝜓) ↔ (𝐷 = 0 → 𝑋 = 𝑌)))
3332imbi2d 329 . . . . 5 (𝑥 = 𝐷 → ((𝜑 → (𝑥 = 0 → 𝜓)) ↔ (𝜑 → (𝐷 = 0 → 𝑋 = 𝑌))))
34 xmulasslem.0 . . . . 5 (𝜑 → (𝑥 = 0 → 𝜓))
3533, 34vtoclg 3239 . . . 4 (𝐷 ∈ ℝ* → (𝜑 → (𝐷 = 0 → 𝑋 = 𝑌)))
361, 35mpcom 37 . . 3 (𝜑 → (𝐷 = 0 → 𝑋 = 𝑌))
37 breq2 4587 . . . . . . 7 (𝑥 = 𝐷 → (0 < 𝑥 ↔ 0 < 𝐷))
3837, 31imbi12d 333 . . . . . 6 (𝑥 = 𝐷 → ((0 < 𝑥𝜓) ↔ (0 < 𝐷𝑋 = 𝑌)))
3938imbi2d 329 . . . . 5 (𝑥 = 𝐷 → ((𝜑 → (0 < 𝑥𝜓)) ↔ (𝜑 → (0 < 𝐷𝑋 = 𝑌))))
4039, 17vtoclga 3245 . . . 4 (𝐷 ∈ ℝ* → (𝜑 → (0 < 𝐷𝑋 = 𝑌)))
411, 40mpcom 37 . . 3 (𝜑 → (0 < 𝐷𝑋 = 𝑌))
4229, 36, 413jaod 1384 . 2 (𝜑 → ((𝐷 < 0 ∨ 𝐷 = 0 ∨ 0 < 𝐷) → 𝑋 = 𝑌))
436, 42mpd 15 1 (𝜑𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3o 1030   = wceq 1475  wcel 1977   class class class wbr 4583   Or wor 4958  0cc0 9815  *cxr 9952   < clt 9953  -𝑒cxne 11819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-xneg 11822
This theorem is referenced by:  xmulass  11989
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