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Mirrors > Home > MPE Home > Th. List > eqord2 | Structured version Visualization version GIF version |
Description: A strictly decreasing real function on a subset of ℝ is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) |
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)) |
Ref | Expression |
---|---|
eqord2 | ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 ↔ 𝑀 = 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltord.1 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
2 | 1 | negeqd 10872 | . . 3 ⊢ (𝑥 = 𝑦 → -𝐴 = -𝐵) |
3 | ltord.2 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) | |
4 | 3 | negeqd 10872 | . . 3 ⊢ (𝑥 = 𝐶 → -𝐴 = -𝑀) |
5 | ltord.3 | . . . 4 ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) | |
6 | 5 | negeqd 10872 | . . 3 ⊢ (𝑥 = 𝐷 → -𝐴 = -𝑁) |
7 | ltord.4 | . . 3 ⊢ 𝑆 ⊆ ℝ | |
8 | ltord.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) | |
9 | 8 | renegcld 11059 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → -𝐴 ∈ ℝ) |
10 | ltord2.6 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴)) | |
11 | 8 | ralrimiva 3180 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ) |
12 | 1 | eleq1d 2895 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℝ ↔ 𝐵 ∈ ℝ)) |
13 | 12 | rspccva 3620 | . . . . . . 7 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝑦 ∈ 𝑆) → 𝐵 ∈ ℝ) |
14 | 11, 13 | sylan 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐵 ∈ ℝ) |
15 | 14 | adantrl 714 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐵 ∈ ℝ) |
16 | 8 | adantrr 715 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐴 ∈ ℝ) |
17 | ltneg 11132 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ -𝐴 < -𝐵)) | |
18 | 15, 16, 17 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐵 < 𝐴 ↔ -𝐴 < -𝐵)) |
19 | 10, 18 | sylibd 241 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → -𝐴 < -𝐵)) |
20 | 2, 4, 6, 7, 9, 19 | eqord1 11160 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 ↔ -𝑀 = -𝑁)) |
21 | 3 | eleq1d 2895 | . . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝑀 ∈ ℝ)) |
22 | 21 | rspccva 3620 | . . . . . 6 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ) |
23 | 11, 22 | sylan 582 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ) |
24 | 23 | adantrr 715 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑀 ∈ ℝ) |
25 | 24 | recnd 10661 | . . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑀 ∈ ℂ) |
26 | 5 | eleq1d 2895 | . . . . . . 7 ⊢ (𝑥 = 𝐷 → (𝐴 ∈ ℝ ↔ 𝑁 ∈ ℝ)) |
27 | 26 | rspccva 3620 | . . . . . 6 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ) |
28 | 11, 27 | sylan 582 | . . . . 5 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ) |
29 | 28 | adantrl 714 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑁 ∈ ℝ) |
30 | 29 | recnd 10661 | . . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑁 ∈ ℂ) |
31 | 25, 30 | neg11ad 10985 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (-𝑀 = -𝑁 ↔ 𝑀 = 𝑁)) |
32 | 20, 31 | bitrd 281 | 1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 ↔ 𝑀 = 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 ⊆ wss 3934 class class class wbr 5057 ℝcr 10528 < clt 10667 -cneg 10863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 |
This theorem is referenced by: basellem4 25653 |
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