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Mirrors > Home > MPE Home > Th. List > 2sqreulem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for 2sqreu 26334 et. (Contributed by AV, 25-Jun-2023.) |
Ref | Expression |
---|---|
2sqreulem4.1 | ⊢ (𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
Ref | Expression |
---|---|
2sqreulem4 | ⊢ ∀𝑎 ∈ ℕ0 ∃*𝑏 ∈ ℕ0 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqreulem3 26331 | . . . 4 ⊢ ((𝑎 ∈ ℕ0 ∧ (𝑏 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0)) → (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) | |
2 | 1 | ralrimivva 3109 | . . 3 ⊢ (𝑎 ∈ ℕ0 → ∀𝑏 ∈ ℕ0 ∀𝑐 ∈ ℕ0 (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) |
3 | 2sqreulem4.1 | . . . . 5 ⊢ (𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | |
4 | 3 | rmobii 3305 | . . . 4 ⊢ (∃*𝑏 ∈ ℕ0 𝜑 ↔ ∃*𝑏 ∈ ℕ0 (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
5 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑏ℕ0 | |
6 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑐ℕ0 | |
7 | nfsbc1v 3711 | . . . . . 6 ⊢ Ⅎ𝑏[𝑐 / 𝑏]𝜓 | |
8 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑏((𝑎↑2) + (𝑐↑2)) = 𝑃 | |
9 | 7, 8 | nfan 1907 | . . . . 5 ⊢ Ⅎ𝑏([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃) |
10 | sbceq1a 3702 | . . . . . 6 ⊢ (𝑏 = 𝑐 → (𝜓 ↔ [𝑐 / 𝑏]𝜓)) | |
11 | oveq1 7217 | . . . . . . . 8 ⊢ (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2)) | |
12 | 11 | oveq2d 7226 | . . . . . . 7 ⊢ (𝑏 = 𝑐 → ((𝑎↑2) + (𝑏↑2)) = ((𝑎↑2) + (𝑐↑2))) |
13 | 12 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑏 = 𝑐 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) |
14 | 10, 13 | anbi12d 634 | . . . . 5 ⊢ (𝑏 = 𝑐 → ((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃))) |
15 | 5, 6, 9, 14 | rmo4f 3645 | . . . 4 ⊢ (∃*𝑏 ∈ ℕ0 (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∀𝑏 ∈ ℕ0 ∀𝑐 ∈ ℕ0 (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) |
16 | 4, 15 | bitri 278 | . . 3 ⊢ (∃*𝑏 ∈ ℕ0 𝜑 ↔ ∀𝑏 ∈ ℕ0 ∀𝑐 ∈ ℕ0 (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) |
17 | 2, 16 | sylibr 237 | . 2 ⊢ (𝑎 ∈ ℕ0 → ∃*𝑏 ∈ ℕ0 𝜑) |
18 | 17 | rgen 3068 | 1 ⊢ ∀𝑎 ∈ ℕ0 ∃*𝑏 ∈ ℕ0 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3058 ∃*wrmo 3061 [wsbc 3691 (class class class)co 7210 + caddc 10729 2c2 11882 ℕ0cn0 12087 ↑cexp 13632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 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 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-nn 11828 df-2 11890 df-n0 12088 df-z 12174 df-uz 12436 df-seq 13572 df-exp 13633 |
This theorem is referenced by: 2sqreunnlem2 26333 2sqreu 26334 2sqreult 26336 2sqreultb 26337 |
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