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| Description: 0lt1 11785 without ax-mulcom 11219. (Contributed by SN, 13-Feb-2024.) | 
| Ref | Expression | 
|---|---|
| sn-0lt1 | ⊢ 0 < 1 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-1ne0 11224 | . . 3 ⊢ 1 ≠ 0 | |
| 2 | 1re 11261 | . . . 4 ⊢ 1 ∈ ℝ | |
| 3 | 0re 11263 | . . . 4 ⊢ 0 ∈ ℝ | |
| 4 | 2, 3 | lttri2i 11375 | . . 3 ⊢ (1 ≠ 0 ↔ (1 < 0 ∨ 0 < 1)) | 
| 5 | 1, 4 | mpbi 230 | . 2 ⊢ (1 < 0 ∨ 0 < 1) | 
| 6 | rernegcl 42401 | . . . . . 6 ⊢ (1 ∈ ℝ → (0 −ℝ 1) ∈ ℝ) | |
| 7 | 2, 6 | mp1i 13 | . . . . 5 ⊢ (1 < 0 → (0 −ℝ 1) ∈ ℝ) | 
| 8 | relt0neg1 42474 | . . . . . . 7 ⊢ (1 ∈ ℝ → (1 < 0 ↔ 0 < (0 −ℝ 1))) | |
| 9 | 2, 8 | ax-mp 5 | . . . . . 6 ⊢ (1 < 0 ↔ 0 < (0 −ℝ 1)) | 
| 10 | 9 | biimpi 216 | . . . . 5 ⊢ (1 < 0 → 0 < (0 −ℝ 1)) | 
| 11 | 7, 7, 10, 10 | mulgt0d 11416 | . . . 4 ⊢ (1 < 0 → 0 < ((0 −ℝ 1) · (0 −ℝ 1))) | 
| 12 | 1red 11262 | . . . . . . 7 ⊢ (1 ∈ ℝ → 1 ∈ ℝ) | |
| 13 | 6, 12 | remulneg2d 42444 | . . . . . 6 ⊢ (1 ∈ ℝ → ((0 −ℝ 1) · (0 −ℝ 1)) = (0 −ℝ ((0 −ℝ 1) · 1))) | 
| 14 | ax-1rid 11225 | . . . . . . . 8 ⊢ ((0 −ℝ 1) ∈ ℝ → ((0 −ℝ 1) · 1) = (0 −ℝ 1)) | |
| 15 | 6, 14 | syl 17 | . . . . . . 7 ⊢ (1 ∈ ℝ → ((0 −ℝ 1) · 1) = (0 −ℝ 1)) | 
| 16 | 15 | oveq2d 7447 | . . . . . 6 ⊢ (1 ∈ ℝ → (0 −ℝ ((0 −ℝ 1) · 1)) = (0 −ℝ (0 −ℝ 1))) | 
| 17 | renegneg 42441 | . . . . . 6 ⊢ (1 ∈ ℝ → (0 −ℝ (0 −ℝ 1)) = 1) | |
| 18 | 13, 16, 17 | 3eqtrd 2781 | . . . . 5 ⊢ (1 ∈ ℝ → ((0 −ℝ 1) · (0 −ℝ 1)) = 1) | 
| 19 | 2, 18 | ax-mp 5 | . . . 4 ⊢ ((0 −ℝ 1) · (0 −ℝ 1)) = 1 | 
| 20 | 11, 19 | breqtrdi 5184 | . . 3 ⊢ (1 < 0 → 0 < 1) | 
| 21 | id 22 | . . 3 ⊢ (0 < 1 → 0 < 1) | |
| 22 | 20, 21 | jaoi 858 | . 2 ⊢ ((1 < 0 ∨ 0 < 1) → 0 < 1) | 
| 23 | 5, 22 | ax-mp 5 | 1 ⊢ 0 < 1 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 −ℝ cresub 42395 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-2 12329 df-3 12330 df-resub 42396 | 
| This theorem is referenced by: sn-ltp1 42494 sn-mulgt1d 42495 reneg1lt0 42496 | 
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