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Mirrors > Home > MPE Home > Th. List > xmulasslem2 | Structured version Visualization version GIF version |
Description: Lemma for xmulass 13326. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmulasslem2 | ⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5152 | . . 3 ⊢ (𝐴 = -∞ → (0 < 𝐴 ↔ 0 < -∞)) | |
2 | 0xr 11306 | . . . . 5 ⊢ 0 ∈ ℝ* | |
3 | nltmnf 13169 | . . . . 5 ⊢ (0 ∈ ℝ* → ¬ 0 < -∞) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ¬ 0 < -∞ |
5 | 4 | pm2.21i 119 | . . 3 ⊢ (0 < -∞ → 𝜑) |
6 | 1, 5 | biimtrdi 253 | . 2 ⊢ (𝐴 = -∞ → (0 < 𝐴 → 𝜑)) |
7 | 6 | impcom 407 | 1 ⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 0cc0 11153 -∞cmnf 11291 ℝ*cxr 11292 < clt 11293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-addrcl 11214 ax-rnegex 11224 ax-cnre 11226 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 |
This theorem is referenced by: xmulgt0 13322 xmulasslem3 13325 |
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