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Mirrors > Home > MPE Home > Th. List > xmulasslem2 | Structured version Visualization version GIF version |
Description: Lemma for xmulass 12681. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmulasslem2 | ⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5070 | . . 3 ⊢ (𝐴 = -∞ → (0 < 𝐴 ↔ 0 < -∞)) | |
2 | 0xr 10688 | . . . . 5 ⊢ 0 ∈ ℝ* | |
3 | nltmnf 12525 | . . . . 5 ⊢ (0 ∈ ℝ* → ¬ 0 < -∞) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ¬ 0 < -∞ |
5 | 4 | pm2.21i 119 | . . 3 ⊢ (0 < -∞ → 𝜑) |
6 | 1, 5 | syl6bi 255 | . 2 ⊢ (𝐴 = -∞ → (0 < 𝐴 → 𝜑)) |
7 | 6 | impcom 410 | 1 ⊢ ((0 < 𝐴 ∧ 𝐴 = -∞) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 0cc0 10537 -∞cmnf 10673 ℝ*cxr 10674 < clt 10675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-addrcl 10598 ax-rnegex 10608 ax-cnre 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 |
This theorem is referenced by: xmulgt0 12677 xmulasslem3 12680 |
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