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Theorem leord2 11505

Description: Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

**Hypotheses**
Ref	Expression
ltord.1	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
ltord.2	⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)
ltord.3	⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)
ltord.4	⊢ 𝑆 ⊆ ℝ
ltord.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
ltord2.6	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴))

**Assertion**
Ref	Expression
leord2	⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ 𝑁 ≤ 𝑀))

Distinct variable groups: 𝑥,𝐵 𝑥,𝑦,𝐶 𝑥,𝐷,𝑦 𝑥,𝑀,𝑦 𝑥,𝑁,𝑦 𝜑,𝑥,𝑦 𝑥,𝑆,𝑦 Allowed substitution hints: 𝐴(𝑥,𝑦) 𝐵(𝑦)

**Proof of Theorem leord2**
Step	Hyp	Ref	Expression
1		ltord.1	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
2	1	negeqd 11215	. . 3 ⊢ (𝑥 = 𝑦 → -𝐴 = -𝐵)
3		ltord.2	. . . 4 ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)
4	3	negeqd 11215	. . 3 ⊢ (𝑥 = 𝐶 → -𝐴 = -𝑀)
5		ltord.3	. . . 4 ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)
6	5	negeqd 11215	. . 3 ⊢ (𝑥 = 𝐷 → -𝐴 = -𝑁)
7		ltord.4	. . 3 ⊢ 𝑆 ⊆ ℝ
8		ltord.5	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
9	8	renegcld 11402	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → -𝐴 ∈ ℝ)
10		ltord2.6	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴))
11	8	ralrimiva 3103	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ)
12	1	eleq1d 2823	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℝ ↔ 𝐵 ∈ ℝ))
13	12	rspccva 3560	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝑦 ∈ 𝑆) → 𝐵 ∈ ℝ)
14	11, 13	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐵 ∈ ℝ)
15	14	adantrl 713	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐵 ∈ ℝ)
16	8	adantrr 714	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐴 ∈ ℝ)
17		ltneg 11475	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ -𝐴 < -𝐵))
18	15, 16, 17	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐵 < 𝐴 ↔ -𝐴 < -𝐵))
19	10, 18	sylibd 238	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → -𝐴 < -𝐵))
20	2, 4, 6, 7, 9, 19	leord1 11502	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ -𝑀 ≤ -𝑁))
21	5	eleq1d 2823	. . . . . 6 ⊢ (𝑥 = 𝐷 → (𝐴 ∈ ℝ ↔ 𝑁 ∈ ℝ))
22	21	rspccva 3560	. . . . 5 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ)
23	11, 22	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ)
24	23	adantrl 713	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑁 ∈ ℝ)
25	3	eleq1d 2823	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝑀 ∈ ℝ))
26	25	rspccva 3560	. . . . 5 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ)
27	11, 26	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ)
28	27	adantrr 714	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑀 ∈ ℝ)
29		leneg 11478	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 ≤ 𝑀 ↔ -𝑀 ≤ -𝑁))
30	24, 28, 29	syl2anc 584	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝑁 ≤ 𝑀 ↔ -𝑀 ≤ -𝑁))
31	20, 30	bitr4d 281	1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ 𝑁 ≤ 𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3887 class class class wbr 5074 ℝcr 10870 < clt 11009 ≤ cle 11010 -cneg 11206

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208

This theorem is referenced by: (None)

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