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Mirrors > Home > MPE Home > Th. List > leagne1 | Structured version Visualization version GIF version |
Description: Deduce inequality from the less-than angle relation. (Contributed by Thierry Arnoux, 25-Feb-2023.) |
Ref | Expression |
---|---|
isleag.p | ⊢ 𝑃 = (Base‘𝐺) |
isleag.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
isleag.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
isleag.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
isleag.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
isleag.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
isleag.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
isleag.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
leagne.1 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉) |
Ref | Expression |
---|---|
leagne1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isleag.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | eqid 2820 | . . 3 ⊢ (Itv‘𝐺) = (Itv‘𝐺) | |
3 | eqid 2820 | . . 3 ⊢ (hlG‘𝐺) = (hlG‘𝐺) | |
4 | isleag.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉)) → 𝐺 ∈ TarskiG) |
6 | isleag.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉)) → 𝐴 ∈ 𝑃) |
8 | isleag.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | 8 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉)) → 𝐵 ∈ 𝑃) |
10 | isleag.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
11 | 10 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉)) → 𝐶 ∈ 𝑃) |
12 | isleag.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
13 | 12 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉)) → 𝐷 ∈ 𝑃) |
14 | isleag.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
15 | 14 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉)) → 𝐸 ∈ 𝑃) |
16 | simplr 767 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉)) → 𝑥 ∈ 𝑃) | |
17 | simprr 771 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉)) → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉) | |
18 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 16, 17 | cgrane1 26596 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉)) → 𝐴 ≠ 𝐵) |
19 | leagne.1 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉) | |
20 | isleag.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
21 | 1, 4, 6, 8, 10, 12, 14, 20 | isleag 26631 | . . 3 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(≤∠‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉))) |
22 | 19, 21 | mpbid 234 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥(inA‘𝐺)〈“𝐷𝐸𝐹”〉 ∧ 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝑥”〉)) |
23 | 18, 22 | r19.29a 3288 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∃wrex 3138 class class class wbr 5059 ‘cfv 6348 〈“cs3 14199 Basecbs 16478 TarskiGcstrkg 26214 Itvcitv 26220 hlGchlg 26384 cgrAccgra 26591 inAcinag 26619 ≤∠cleag 26620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 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 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-fzo 13031 df-hash 13688 df-word 13859 df-concat 13918 df-s1 13945 df-s2 14205 df-s3 14206 df-trkgc 26232 df-trkgcb 26234 df-trkg 26237 df-cgrg 26295 df-hlg 26385 df-cgra 26592 df-leag 26630 |
This theorem is referenced by: (None) |
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