enqeq - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > enqeq		Structured version Visualization version GIF version

Theorem enqeq 10852

Description: Corollary of nqereu 10847: if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)

**Assertion**
Ref	Expression
enqeq	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐴 = 𝐵)

Proof of Theorem enqeq Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpa 1149	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
2		elpqn 10843	. . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
3	2	3ad2ant2 1135	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ∈ (N × N))
4		nqereu 10847	. . . 4 ⊢ (𝐵 ∈ (N × N) → ∃!𝑥 ∈ Q 𝑥 ~_Q 𝐵)
5		reurmo 3346	. . . 4 ⊢ (∃!𝑥 ∈ Q 𝑥 ~_Q 𝐵 → ∃*𝑥 ∈ Q 𝑥 ~_Q 𝐵)
6	3, 4, 5	3syl 18	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → ∃*𝑥 ∈ Q 𝑥 ~_Q 𝐵)
7		df-rmo 3343	. . 3 ⊢ (∃𝑥 ∈ Q 𝑥 ~_Q 𝐵 ↔ ∃𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵))
8	6, 7	sylib 218	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵))
9		3simpb 1150	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵))
10		simp2 1138	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ∈ Q)
11		enqer 10839	. . . . 5 ⊢ ~_Q Er (N × N)
12	11	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → ~_Q Er (N × N))
13	12, 3	erref 8659	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐵 ~_Q 𝐵)
14	10, 13	jca 511	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵))
15		eleq1 2825	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Q ↔ 𝐴 ∈ Q))
16		breq1 5089	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ~_Q 𝐵 ↔ 𝐴 ~_Q 𝐵))
17	15, 16	anbi12d 633	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵) ↔ (𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵)))
18		eleq1 2825	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ Q ↔ 𝐵 ∈ Q))
19		breq1 5089	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ~_Q 𝐵 ↔ 𝐵 ~_Q 𝐵))
20	18, 19	anbi12d 633	. . 3 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵) ↔ (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵)))
21	17, 20	moi 3665	. 2 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~_Q 𝐵) ∧ ((𝐴 ∈ Q ∧ 𝐴 ~_Q 𝐵) ∧ (𝐵 ∈ Q ∧ 𝐵 ~_Q 𝐵))) → 𝐴 = 𝐵)
22	1, 8, 9, 14, 21	syl112anc 1377	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~_Q 𝐵) → 𝐴 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∃*wmo 2538 ∃!wreu 3341 ∃*wrmo 3342 class class class wbr 5086 × cxp 5624 Er wer 8635 Ncnpi 10762 ~_Q ceq 10769 Qcnq 10770

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 2709 ax-sep 5232 ax-nul 5242 ax-pr 5372 ax-un 7684

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-oadd 8404 df-omul 8405 df-er 8638 df-ni 10790 df-mi 10792 df-lti 10793 df-enq 10829 df-nq 10830

This theorem is referenced by: nqereq 10853 ltsonq 10887

W3C validator