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Theorem leord2 11680

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 11387	. . 3 ⊢ (𝑥 = 𝑦 → -𝐴 = -𝐵)
3		ltord.2	. . . 4 ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)
4	3	negeqd 11387	. . 3 ⊢ (𝑥 = 𝐶 → -𝐴 = -𝑀)
5		ltord.3	. . . 4 ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)
6	5	negeqd 11387	. . 3 ⊢ (𝑥 = 𝐷 → -𝐴 = -𝑁)
7		ltord.4	. . 3 ⊢ 𝑆 ⊆ ℝ
8		ltord.5	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)
9	8	renegcld 11577	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → -𝐴 ∈ ℝ)
10		ltord2.6	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴))
11	8	ralrimiva 3129	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ)
12	1	eleq1d 2821	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℝ ↔ 𝐵 ∈ ℝ))
13	12	rspccva 3563	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝑦 ∈ 𝑆) → 𝐵 ∈ ℝ)
14	11, 13	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐵 ∈ ℝ)
15	14	adantrl 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐵 ∈ ℝ)
16	8	adantrr 718	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐴 ∈ ℝ)
17		ltneg 11650	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ -𝐴 < -𝐵))
18	15, 16, 17	syl2anc 585	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐵 < 𝐴 ↔ -𝐴 < -𝐵))
19	10, 18	sylibd 239	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → -𝐴 < -𝐵))
20	2, 4, 6, 7, 9, 19	leord1 11677	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ -𝑀 ≤ -𝑁))
21	5	eleq1d 2821	. . . . . 6 ⊢ (𝑥 = 𝐷 → (𝐴 ∈ ℝ ↔ 𝑁 ∈ ℝ))
22	21	rspccva 3563	. . . . 5 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ)
23	11, 22	sylan 581	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ)
24	23	adantrl 717	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑁 ∈ ℝ)
25	3	eleq1d 2821	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝑀 ∈ ℝ))
26	25	rspccva 3563	. . . . 5 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ)
27	11, 26	sylan 581	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ)
28	27	adantrr 718	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑀 ∈ ℝ)
29		leneg 11653	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 ≤ 𝑀 ↔ -𝑀 ≤ -𝑁))
30	24, 28, 29	syl2anc 585	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝑁 ≤ 𝑀 ↔ -𝑀 ≤ -𝑁))
31	20, 30	bitr4d 282	1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ 𝑁 ≤ 𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ⊆ wss 3889 class class class wbr 5085 ℝcr 11037 < clt 11179 ≤ cle 11180 -cneg 11378

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380

This theorem is referenced by: (None)

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