Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltadd12dd | Structured version Visualization version GIF version |
Description: Addition to both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
ltadd12dd.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltadd12dd.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltadd12dd.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
ltadd12dd.d | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
ltadd12dd.ac | ⊢ (𝜑 → 𝐴 < 𝐶) |
ltadd12dd.bd | ⊢ (𝜑 → 𝐵 < 𝐷) |
Ref | Expression |
---|---|
ltadd12dd | ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltadd12dd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltadd12dd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 1, 2 | readdcld 10672 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
4 | ltadd12dd.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 4, 2 | readdcld 10672 | . 2 ⊢ (𝜑 → (𝐶 + 𝐵) ∈ ℝ) |
6 | ltadd12dd.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
7 | 4, 6 | readdcld 10672 | . 2 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ) |
8 | ltadd12dd.ac | . . 3 ⊢ (𝜑 → 𝐴 < 𝐶) | |
9 | 1, 4, 2, 8 | ltadd1dd 11253 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐵)) |
10 | ltadd12dd.bd | . . 3 ⊢ (𝜑 → 𝐵 < 𝐷) | |
11 | 2, 6, 4, 10 | ltadd2dd 10801 | . 2 ⊢ (𝜑 → (𝐶 + 𝐵) < (𝐶 + 𝐷)) |
12 | 3, 5, 7, 9, 11 | lttrd 10803 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 + caddc 10542 < clt 10677 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 |
This theorem is referenced by: sge0xaddlem1 42722 smfaddlem1 43046 |
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